logic, algebra and topology · 2010-11-15 · logic, algebra and topology investigations into...

Logic, Algebra and Topology

Investigations into canonical extensions, duality theory andpoint-free topology

Jacob Vosmaer

ILLC Dissertation Series DS-2010-10

For further information about ILLC-publications, please contact

Institute for Logic, Language and ComputationUniversiteit van Amsterdam

Science Park 9041098 XH Amsterdam

phone: +31-20-525 6051fax: +31-20-525 5206e-mail: [email protected]

homepage: http://www.illc.uva.nl/



Academisch Proefschrift

ter verkrijging van de graad van doctor aan deUniversiteit van Amsterdam

op gezag van de Rector Magnificusprof. dr. D.C. van den Boom

ten overstaan van een door het college voorpromoties ingestelde commissie, in het openbaar

te verdedigen in de Agnietenkapelop dinsdag 14 december 2010, te 14.00 uur

door

Jacob Vosmaer

geboren te Amsterdam.

Promotiecommissie

Promotores: Prof. dr. M. GehrkeProf. dr. Y. Venema

Overige leden: Prof. dr. J.F.A.K. van BenthemDr. G. BezhanishviliProf. dr. D.H.J. de JonghProf. dr. A. JungDr. A. PalmigianoProf. dr. H.A. Priestley

Faculteit der Natuurwetenschappen, Wiskunde en InformaticaUniversiteit van AmsterdamScience Park 9041098 XH Amsterdam

This research was supported by the Netherlands Organization for Scientific Re-search (NWO) VICI grant 639.073.501.

Copyright c© 2010 by Jacob Vosmaer

Printed and bound by Ipskamp Drukkers.

ISBN: 978–90–5776–214–7

voor Liesbeth en Therus

v

Contents

Acknowledgments xi

1 Introduction 11.1 Logic, algebra and topology . . . . . . . . . . . . . . . . . . . . . 11.2 Mathematical surroundings . . . . . . . . . . . . . . . . . . . . . 11.3 Survey of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Canonical extensions: a domain theoretic approach 92.1 Canonical extension via filters, ideals and topology . . . . . . . . 10

2.1.1 The canonical extension of a lattice . . . . . . . . . . . . . 112.1.2 Topologies on posets and completions . . . . . . . . . . . . 152.1.3 Characterizing the canonical extension via the δ-topology . 182.1.4 Basic properties of the canonical extension . . . . . . . . . 242.1.5 Conclusions and further work . . . . . . . . . . . . . . . . 27

2.2 Canonical extensions of maps I: order-preserving maps . . . . . . 272.2.1 The lower and upper extensions of an order-preserving map 282.2.2 Operators and join-preserving maps . . . . . . . . . . . . . 332.2.3 Canonical extension as a functor I: lattices only . . . . . . 422.2.4 Conclusions and further work . . . . . . . . . . . . . . . . 44

2.3 Canonical extensions via dcpo presentations . . . . . . . . . . . . 452.3.1 Dcpo presentations . . . . . . . . . . . . . . . . . . . . . . 472.3.2 A dcpo presentation of the canonical extension . . . . . . . 482.3.3 Extending maps via dcpo presentations . . . . . . . . . . . 502.3.4 Conclusions and further work . . . . . . . . . . . . . . . . 51

3 Canonical extensions and topological algebra 533.1 Topological algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.1.1 Compact Hausdorff algebras . . . . . . . . . . . . . . . . . 553.1.2 Profinite algebras and profinite completions . . . . . . . . 56

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3.1.3 Topological lattices . . . . . . . . . . . . . . . . . . . . . . 633.2 Canonical extensions of maps II: maps into profinite lattices . . . 68

3.2.1 Extending maps via lim inf and lim sup . . . . . . . . . . . 693.2.2 Maps into profinite lattices . . . . . . . . . . . . . . . . . . 743.2.3 Canonical extension and function composition . . . . . . . 783.2.4 Conclusions and further work . . . . . . . . . . . . . . . . 82

3.3 Canonical extension as a functor II: lattice-based algebras . . . . 833.3.1 Order types and canonical extension types . . . . . . . . . 843.3.2 Preservation of homomorphisms . . . . . . . . . . . . . . . 873.3.3 Canonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . 893.3.4 Conclusions and further work . . . . . . . . . . . . . . . . 93

3.4 Profinite completion and canonical extension . . . . . . . . . . . . 933.4.1 Universal properties of canonical extension . . . . . . . . . 943.4.2 Finitely generated varieties . . . . . . . . . . . . . . . . . . 993.4.3 Canonical extension and monotone topological algebras . . 1053.4.4 Conclusions and further work . . . . . . . . . . . . . . . . 109

4 Duality, profiniteness and completions 1114.1 Dualities for distributive lattices with operators . . . . . . . . . . 112

4.1.1 Semi-topological DLO’s . . . . . . . . . . . . . . . . . . . 1134.1.2 Colimits of ordered Kripke frames . . . . . . . . . . . . . . 1164.1.3 Duality for profinite DLO’s . . . . . . . . . . . . . . . . . 1204.1.4 Profinite completion via duality . . . . . . . . . . . . . . . 1224.1.5 Conclusions and further work . . . . . . . . . . . . . . . . 127

4.2 A brief survey of subcategories of DLO . . . . . . . . . . . . . . . 1274.2.1 Distributive lattices . . . . . . . . . . . . . . . . . . . . . . 1284.2.2 Boolean algebras . . . . . . . . . . . . . . . . . . . . . . . 1294.2.3 Heyting algebras . . . . . . . . . . . . . . . . . . . . . . . 130

4.3 Duality for topological Boolean algebras with operators . . . . . . 1324.3.1 Duality for Boolean topological BAO’s . . . . . . . . . . . 1334.3.2 Ultrafilter extensions of image-finite Kripke frames . . . . 1374.3.3 Conclusions and further work . . . . . . . . . . . . . . . . 139

5 Coalgebraic modal logic in point-free topology 1415.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

5.2.1 Basic mathematics . . . . . . . . . . . . . . . . . . . . . . 1445.2.2 Category theory . . . . . . . . . . . . . . . . . . . . . . . . 1455.2.3 Relation lifting . . . . . . . . . . . . . . . . . . . . . . . . 1475.2.4 Frames and their presentations . . . . . . . . . . . . . . . 1525.2.5 Powerlocales via and ♦ . . . . . . . . . . . . . . . . . . 154

5.3 The T -powerlocale construction . . . . . . . . . . . . . . . . . . . 1575.3.1 Introducing the T -powerlocale . . . . . . . . . . . . . . . . 157

viii

5.3.2 Basic properties of the T -powerlocale . . . . . . . . . . . . 161

5.3.3 Two examples of the T -powerlocale construction . . . . . . 165

5.3.4 Categorical properties of the T -powerlocale . . . . . . . . . 171

5.3.5 T -powerlocales via flat sites . . . . . . . . . . . . . . . . . 179

5.4 Preservation results . . . . . . . . . . . . . . . . . . . . . . . . . . 185

5.4.1 Regularity and zero-dimensionality . . . . . . . . . . . . . 185

5.4.2 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . 191

5.5 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

A Preliminaries 199

A.1 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

A.2 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

A.2.1 Categories and functors . . . . . . . . . . . . . . . . . . . 200

A.2.2 Adjunctions of categories . . . . . . . . . . . . . . . . . . . 202

A.3 Order theory and domain theory . . . . . . . . . . . . . . . . . . . 202

A.3.1 Pre-orders and partial orders . . . . . . . . . . . . . . . . . 202

A.3.2 Adjunctions of partially ordered sets . . . . . . . . . . . . 204

A.3.3 Dcpo’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

A.3.4 Order and topology . . . . . . . . . . . . . . . . . . . . . . 205

A.4 Lattice theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

A.4.1 Semilattices and suplattices . . . . . . . . . . . . . . . . . 205

A.4.2 Lattices and complete lattices . . . . . . . . . . . . . . . . 206

A.4.3 Distributive lattices, Heyting algebras and Boolean algebras 206

A.5 Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

A.5.1 The ideal (and filter) completion of a pre-order . . . . . . 207

A.5.2 The MacNeille completion of a pre-order . . . . . . . . . . 209

A.6 Universal algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

A.6.1 Ω-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

A.6.2 Homomorphic images, subalgebras and products . . . . . . 210

A.6.3 Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

A.6.4 Terms and equations . . . . . . . . . . . . . . . . . . . . . 214

A.7 General topology . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

A.7.1 Topological spaces . . . . . . . . . . . . . . . . . . . . . . 214

A.7.2 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

A.7.3 Separation and compactness . . . . . . . . . . . . . . . . . 216

A.8 Duality for ordered Kripke frames . . . . . . . . . . . . . . . . . . 217

Bibliography 219

Index 227

Samenvatting 231

ix

Abstract 233

x

Acknowledgments

In this section I would like to acknowledge the difference that certain people madefor me while I was doing my PhD research.

Scientific contacts For starters, I would like to thank my main supervisors(promotores) Mai Gehrke and Yde Venema, and the Netherlands Organisationfor Scientific Research (NWO). The research reported on in this dissertation wascarried out at the Institute for Logic, Language and Computation of the Universityof Amsterdam with NWO funding, as part of the VICI ‘Algebra and Coalgebra’project of which Yde Venema is the principal investigator. For four years, startingin November 2006, I have been able to benefit from his supervision and mentorship.In my first year Alessandra Palmigiano served as my secondary supervisor, forwhich I am also very grateful. During my second year, Yde attracted Mai Gehrkeas my second main supervisor. Since Mai is based at the Radboud University inNijmegen, my interactions with her have been slightly less frequent than thosewith Yde, but on every occasion it was both a great pleasure and a great help tosee her. Time and again, Mai and Yde have shown great concern not only for mysuccess as a young academic, but also for my general wellbeing. This has meant agreat deal to me.

I would also like to thank my co-authors for selected papers, namely GuramBezanishvili, Mai Gehrke, Yde Venema and Steve Vickers. My first publicationin a peer-reviewed journal [16] was a joint publication with Guram Bezanishvili.I learned a lot from working with him and I am grateful for the high standardshe demanded from both of us. The two papers I co-authored with Mai Gehrkewere written in a breeze; I enjoyed working on them very much. Finally, I ampleased to have been collaborating with Yde Venema and Steve Vickers on an asyet unpublished paper, on which Chapter 5 is based.

While I was working on my PhD during the last four years, my immediateresearch environment consisted of Yde’s Algebra & Coalgebra group and its‘extended family’, which is grounded in the University of Amsterdam but extends

xi

far beyond it. Many members of this family are mentioned elsewhere in theseacknowledgments; of those who are not, I would like to mention Nick Bezhanishvili,Vincenzo Ciancia, Dion Coumans, Sam van Gool, Helle Hansen, Christian Kissig,Clemens Kupke, Loes Olde Loohuis and Alexandra Silva.

My work also benefited greatly from several visits to other universities I madeduring the last four years. In early 2008, I visited Hilary Priestley at the Universityof Oxford. I enjoyed my discussions with her a great deal. In late 2009, I visitedSteve Vickers and Achim Jung at the University of Birmingham. This was anothervery nice opportunity for mathematical discussions, and it led to the collaborationwith Steve and Yde on which Chapter 5 is based. Finally I would like to mentionmy visit to Mai Gehrke and Paul-Andre Mellies at Universite Paris 7 – DenisDiderot. I enjoyed having the opportunity to give a very technical and interactivetalk in Paul-Andre’s Semantics Seminar.

At this stage, I have already mentioned most members of my PhD committee,which, besides of Mai and Yde, consists of Johan van Benthem, Guram Bezhan-ishvili, Dick de Jongh, Achim Jung, Alessandra Palmigiano and Hilary Priestley.They took upon them the task to critically judge my work, for which I am mostgrateful. Their interest and efforts mean a lot to me. I feel fortunate knowingthat this dissertation has been subjected to their scrutiny. I am also grateful toGuram, Hilary, Mai and Yde for pointing out many typos in the manuscript; Iassume full responsibility for all that remain.

Concluding this brief discussion of scientific contacts, I would like to acknowl-edge the discussions I had with people I met at OAL 2007 in Nashville, TANCL2007 in Oxford, BLAST 2008 in Denver, AiML 2008 in Nancy, MLNL 2009 inNijmegen, TACL 2009 in Amsterdam, the Coalgebraic Logics workshop in 2009 atDagstuhl, ALCOP 2010 in London and MLNL 2010 in Utrecht. I will not try tolist all the people I have enjoyed speaking with at these various occasions; I wouldlike to single out Rob Goldblatt and Alexander Kurz because of the insightfuldiscussions I have had with them.

University contacts While I conducted my PhD research I was based at theInstitute for Logic, Language and Computation (ILLC) at the University ofAmsterdam (UvA). The ILLC is run by managing director Ingrid van Loon anda secretarial staff which, during my time at the institute, has consisted of KarinGigengack, Marjan Veldhuisen, Peter van Ormondt and Tanja Kassenaar. Thesepeople have made numerous positive contributions to my work environment atthe UvA, for which I am most grateful. At this point I should also mention ReneGoedman, the receptionist at the Euclides building, who was really a part of ourinstitute during the time it was based at said building.

Naturally, the scientific staff, post-docs and PhD students of the ILLC werealso an important part of my work environment. Out of all these people, I wouldlike to explicitly mention my office mates Fernando Velazquez Quesada, Gaelle

xii

Fontaine, Levan Uridia and Raul Andres Leal, who were always open to havingdiscussions.

Concluding this very brief overview of university contacts, I would like tothank the secretarial staff of the ILLC for organizing the weekly Monday morningcoffee break, and scientific director Leen Torenvliet, who organized the institute’sweekly Friday afternoon drinks together with Peter van Ormondt.

Other contacts To conclude these acknowledgements, I would like to mentionsome people who provided healthy distractions from my research, thus contributingto it. The following people distracted me at the university: Gaelle, Olivia, Reutand Teju. Other distractions were provided by Bert-Jan, Jael, Lodewijk, Nynkeand Rembert. In addition, Nila provided a writer’s retreat during the final stagesof writing my dissertation. I am grateful to all these friends.

For a long time, my rowing endeavours at my rowing club A.S.R. Nereussuccesfully diverted my attention away from research. Of all the people there whomade rowing such a wonderful thing to spend my time on, I would like to singleout my coach Roel Luijnenburg, who has been a source of invaluable advice.

Finally, I would like to thank my siblings Sara and Arend, my aunt MartineVosmaer and my parents Liesbeth and Therus Vosmaer-de Bruin for supportingme.

Amsterdam, November 2010

xiii

Chapter 1

Introduction

1.1 Logic, algebra and topology

The connection between logic, algebra and topology is intricate and well-known.The archetypical examples of this connection are provided by the works of G. Booleand M.H. Stone. In the 1840s, Boole, one of the founding fathers of modern logic,made his ‘laws of thought’ the subject of mathematical investigation by treatinglogic as if it were algebra [21]. Boole’s algebraic analysis of logic led to the studyof Boolean algebras. In the 1930s, Stone showed that Boolean algebras can befaithfully represented through using topology as (the duals of) Stone spaces, whichwas a revolutionary contribution to both algebra and topology [81]. Out of thesetwo ideas, the algebraization of logic and duality between algebra and topology,many research fields have sprung. In this dissertation, we present mathematicalinvestigations which contribute to some of the research areas of logic, mathematicsand computer science which developed out of the combined work of Boole andStone: canonical extensions, extended Stone duality and point-free topology.

1.2 Mathematical surroundings

We will begin by sketching some of the mathematical context in which our workmay be viewed.

Stone duality

Boolean algebras are the mathematical models Boole used to study ‘the laws ofthought’, or classical propositional logic as we would call it nowadays. Stone’sfamous duality theorem for Boolean algebras is a two-edged sword. One edgeamounts to the fact that any Boolean algebra B can be represented as the algebraof closed-and-open (clopen) subsets of a topological space X: B ' X∗, where (·)∗is the function that assigns to a topological space X its algebra of clopen sets,

1

2 Chapter 1. Introduction

with union, intersection and complementation as its algebra operations. Morespecifically, what Stone showed is that given B, one can define a topological spaceB∗ such that B ' (B∗)∗. In addition, he showed that such a space B∗ is alwayscompact, Hausdorff and zero-dimensional (recall that a space X is zero-dimensionalif the clopen subsets of X form a basis for its topology). We call such spacesBoolean spaces or Stone spaces; in this dissertation we will use the former term.The other edge of the sword is now that if X is a Boolean space, then X ' (X∗)∗.

The constructions (·)∗ and (·)∗ are not merely defined on Boolean algebrasand Boolean spaces: they are also defined on functions. Specifically, if f : A→ Bis a Boolean algebra homomorphism, then f∗ : B∗ → A∗ is a continuous function,and conversely, if g : X → Y is a continuous function between Boolean spacesthen g∗ : Y ∗ → X∗ is a Boolean algebra homomorphism. These assignments arecontravariant functors, meaning the directions of the arrows are reversed. For thisreason, the equivalence between Boolean algebras and Boolean spaces is called adual equivalence or a duality of categories. Stone duality has proven to be a veryinfluential mathematical discovery, see e.g. [54, Introduction]. Two research areaswhich are based on Stone duality play a central role in this dissertation: relationalsemantics for modal logic and point-free topology.

Relational semantics for modal logic

For our purposes, modal logic consists of propositional logic enriched with modalconnectives, often denoted , ♦, meant to express modalities such as ‘possibly ϕ’,‘agent a knows that ϕ’, ‘at some time in the future, ϕ’, ‘ϕ holds with probability p’,etc. A wide class of modal logics, known as normal modal logics, admit relationalsemantics involving Kripke frames [19]. Kripke frames are structures consisting ofa set X enriched with finitary relations R ⊆ Xn+1, for each modality involved.

Extended Stone duality

Stone duality for Boolean algebras can be extended to a representation theory formodal algebras, the algebraic counterparts of normal modal logics. This extendedduality, known as Jonsson-Tarski duality [58], shows how one can give any modalalgebra a representation as a topological relational structure. Many properties ofmodal logics can be understood in terms of Jonsson-Tarksi duality; however, ifwe want to also involve Kripke frames, i.e. discrete topological structures, thenwe run into a second duality, usually referred to as discrete duality. Discreteduality for modal algebras is based on the well-known duality between completeatomic Boolean algebras and sets [83], which in its turn is based on the fact thata complete atomic algebra can be uniquely described in terms of its set of atoms.The duality between complete atomic Boolean algebras and sets can be extendedinto a duality between complete, atomic, completely additive modal algebras(perfect modal algebras) and Kripke frames.

1.2. Mathematical surroundings 3

The connections between these two dualities are sketched in Figure 1.1. Thehorizontal connections are the dualities: the functors (·)∗ and (·)∗ for Jonsson-Tarski duality, and (·)+ and (·)+ for the discrete duality. The vertical connectionsin the middle are in themselves rather innocuous: on the left, we have the inclusionof the category of perfect modal algebras into the category of all modal algebras;this inclusion exists because any perfect modal algebra is a fortiori a modal algebra.On the right, we have the forgetful functor U , which takes a topological relationalstructure and strips it of its topology, yielding a bare, discrete relational structure.

logicalcalculiOO

OOO

modalalgebras

(·)∗ // descriptivegeneral frames

(·)∗oo

U

perfect

modal algebras

⊆

(·)+ // Kripkeframes

(·)+oo

relationalsemantics

OOOOO

Figure 1.1: The double duality diagram for modal algebras and Kripke frames

In the upper left corner of Figure 1.1, we have logical calculi consisting ofHilbert systems, Gentzen calculi, natural deduction, etc. for modal logics. Thesquigly arrow connecting the logical calculi with modal algebras is covered byalgebraic logic [20], which tells us how to capture logical calculi in systems ofalgebras. The squigly arrow in the bottom right corner, connecting Kripke framesand relational semantics, is covered by the model theory of modal logic, whichtells us how to interpret modal formulas using Kripke frames [19]. The study ofthe logical properties of normal modal logics can now be understood as a study ofthe connections through the square between the logical calculi in the upper leftand the relational semantics in the lower right.

Canonical extensions

The vertical connection on the left side of Figure 1.1 is the subject of study in thetheory of canonical extensions [58]. Canonical extensions are a construction thatallows one to pass from modal algebras to perfect modal algebras, thus adding anew arrow to the diagram.

If one takes a modal algebra A from the upper left corner and ‘pulls it aroundclockwise’, one obtains a perfect modal algebra (UA∗)+, into which A can be


embedded in a natural way. It turns out that the embedding of A into (UA∗)+

can be described in purely abstract terms as an embedding e : A→ Aδ of A intoa perfect modal algebra Aδ, the canonical extension of A, without even assumingthe existence of the two dualities in Figure 1.1. This latter point can be significantbecause Jonsson-Tarski duality is dependent on the Axiom of Choice, a featureit inherits from Stone duality. Certain properties of modal logics can now beunderstood as properties of canonical extensions of modal algebras. An addedadvantage of this stems from the fact that canonical extensions can be definedfor algebras associated with many other logics besides modal logic. This makescanonical extensions into a tool for a generalized, uniform investigation of logicalproperties of a wide range of logics.

Coalgebraic modal logic

In coalgebraic modal logic, a somewhat different view on the double dualitydiagram of Figure 1.1 is adopted. One separates modal logic into a Booleanbase and a modal ‘add-on’, and similarly, Kripke frames are separated into theirunderlying set of states and a transition type. In the case of Kripke frames,the transition type corresponds to the relations on the frame which are usedto interpret modal connectives. Technically, this is made precise by describingabstract transition systems as coalgebras and specifying two functors. One onBoolean algebras, describing the supra-Boolean logical structure, and one on setsdescribing the transition type of the coalgebras involved, resulting in a diagramas in Figure 1.2. In the case of basic modal logic, T is the powerset functor andL is the free ∧-semilattice functor. In coalgebraic logic, one now studies modal

BAL++

U(·)∗33 Set

(·)+ss

Tss

Figure 1.2: The duality diagram of coalgebraic logic

logics (specified by the functor L) in relation to coalgebras (specified by T ), seenas abstract transition systems.

Another coalgebraic interpretation of duality for modal algebras can be foundin the fact that if one considers the free ∧-semilattice functor M , which allows oneto see modal algebras as M -algebras over Boolean algebras, then one can showthat M dually corresponds to the Vietoris hyperspace functor V on Boolean spaces.Descriptive general frames can then be seen as V -coalgebras in the category ofBoolean spaces.

1.3. Survey of Contents 5

Point-free topology

Although Stone called his duality theorem a representation theorem for Booleanalgebras, one might as well see it as a representation theorem for Boolean spaces.When dealing with a problem involving Boolean spaces, one can translate it usingStone duality into a problem involving Boolean algebras and then use algebraicrather than topological methods to analyze the problem. Moreover, at any stageof the analysis, one can switch back to looking at Boolean spaces. In point-freetopology [54], this duality-based view is taken as the primary way to look attopology. This approach consists of two conceptual ingredients. The first is togeneralize Stone duality so that it encompasses all topological spaces, at the priceof weakening the dual equivalence of Stone duality for Boolean spaces to a dualadjunction between topological spaces and frames, the algebras that arise whenone views Stone duality at this generalized level. Frames are complete lattices inwhich infinite joins distribute over finite meets; a frame homomorphism is a mapwhich preserves finite meets and infinite joins. Just as Boolean algebras can beseen as models for classical propositional logic, frames can be seen as models forgeometric propositional logic, which is a logic with finite conjunctions and infinitedisjunctions. The second conceptual ingredient in the framework of point-freetopology is to view frames as ‘point-free spaces’, which we call locales, rather thanas algebras. In this dissertation we will take a predominantly algebraic approachto point-free topology however, meaning that we will mostly deal with frames.

1.3 Survey of Contents

We will now give a broad overview of the contents of this dissertation.

Canonical extensions

The framework of canonical extensions as we present it consists of two components:a construction on lattices, which is a lattice completion, and a construction forextending maps between lattices. Canonical extensions of maps are used bothto define the canonical extensions of a given lattice-based algebra A, since theoperations on A are maps between powers of A, and to define extensions ofhomomorphisms between lattice-based algebras f : A → B. In our discussionof canonical extensions in Chapters 2 and 3, our focus is on three subjects:topological and categorical properties of canonical extensions, and the view oncanonical extensions as dcpo algebras.

Regarding the topological properties of canonical extensions, we would like topoint out the topological characterization of the canonical extension of a latticein §2.1.3, our broad discussion of topological properties of canonical extensions ofmaps, beyond what was previously known, in §2.2 and §3.2, and our investiagations


into universal properties of canonical extensions with respect to topological lattice-based algebras in §3.4. This brings us to the categorical properties of canonicalextensions. One of the crucial facts behind these universal properties is thefact, established in §3.3, that canonical extensions preserve surjective algebrahomomorphisms, a fact which was previously only known to hold under assumptionof distributivity.

In our discussion of canonical extensions of order-preserving maps in §2.2, theideal and filter completion functors play an important role. They reprise this rolewhen we look at canonical extensions as dcpo algebras in §2.3 and §3.3.3. Here weshow how canonicity results for distributive lattices with operators (DLO’s) canbe understood using more general results about dcpo algebras.

Duality for topological lattice-based algebras

When studying distributive lattices with operators (or DLO’s for short), a class ofalgebras which arises naturally in algebraic logic, one has two categorical dualitiesat one’s disposal. The first one is the extended Priestley duality between ‘plain’DLO’s and topological ordered Kripke frames, known as relational Priestley spaces.The second one is the discrete duality between so-called perfect DLO’s and orderedKripke frames sans topology. In our discussion of duality for DLO’s in Chapter4, we propose that the proper perspective on perfect DLO’s is to regard them assemi-topological DLOs. We will use discrete duality for semi-topological DLO’sto provide a number of dual characterization results. In §4.1, we characterizeprofinite DLO’s as the duals of hereditarily finite ordered Kripke frames, andaccordingly we dually characterize the profinite completion of a DLO A relative tothe prime filter frame of A. In §4.2, we briefly discuss how these results specializeto distributive lattices, Boolean algebras and Heyting algebras. Finally, in §4.3,we characterize compact Hausdorff Boolean algebras with operators as the duals ofimage-finite Kripke frames, and we use this duality to study ultrafilter extensionsof Kripke frames using duality.

Powerlocales and coalgebraic logic

In Chapter 5 we use techniques from coalgebraic logic to describe and generalizethe Vietoris powerlocale construction from point-free topology. The idea is thefollowing: we take an axiomatization of coalgebraic logics for T -coalgebras, where Tis an arbitrary weak pullback-preserving standard set functor, known as the Cariocaaxiomatization, and we then use this axiomatization to algebraically describe anew construction on locales, the T -powerlocale construction in §5.3. The usualVietoris powerlocale is now the Pω-powerlocale, where Pω is the finite powersetfunctor. We then proceed to prove a number of properties of T -powerlocales. Forinstance, we show in §5.3.5 that T -powerlocales admit a flat site presentation,which is a technical property that allows us to disentangle the roles of conjunctions

1.3. Survey of Contents 7

and disjunctions. Moreover, in §5.4 we show that the T -powerlocale constructionpreserves (point-free) topological properties such as regularity and the combinationof compactness and zero-dimensionality.

Acknowledgments

Chapters 2 and 3 contain results from joint work with M. Gehrke [42, 43]. Chapter5 is based on joint work with S. Vickers and Y. Venema.

Chapter 2

Canonical extensions: a domaintheoretic approach

In this chapter, we provide a perspective on the toolkit of canonical extensionsof lattices and order-preserving maps with an eye to connections with domaintheory. This leads to new results and new insights into the reasons why and whencanonical extensions work.

Canonical extensions were introduced by Jonsson and Tarski in 1951 [58, 59],as part of the representation theory for Boolean algebras with operators (BAOs),which play an important role in algebraic logic. Examples of BAOs consideredby Jonsson & Tarski were e.g. relation algebras, cylindric algebras and closurealgebras. Another example, which was not immediately recognized, is the class ofmodal algebras, which are used to study modal logic via algebra. The connectionbetween BAOs and modal logic, or rather the connection between BAOs andKripke semantics for modal logic via Stone duality became more widely recognizedin the 1970s via the work of Thomason [87] and Goldblatt [46]; see [48] for historicalcontext. The algebraic approach to BAO representation theory via the purelyalgebraic theory of canonical extensions was revived in the 1990s and 2000s in thework of Jonsson, Gehrke and Harding [57, 38, 34, 39] and De Rijke and Venema[31], and this approach has remained active since. Moreover canonical extensionshave been generalized from a representation theory for Boolean algebras withoperators to a more general representation theory toolkit for lattice-based algebras.We will wait with defining canonical extensions of lattice-based algebras untilChapter 3. In this chapter, we will introduce the reader to two important parts ofthe canonical extensions toolkit: canonical extensions of lattices and canonicalextensions of order-preserving maps between lattices. Moreover, while doing sowe will demonstrate the utility of methods from domain theory in relation tocanonical extensions.

Domain theory was pioneered by D.S. Scott, with the aim to “give a mathe-matical semantics for high-level computer languages” [80]. It has been used tostudy subjects as diverse as recursive equations, semantics for untyped λ-calculus,

9

10 Chapter 2. Canonical extensions: a domain theoretic approach

computability and partial information (see [1]). In this chapter we will borrowseveral techniques from the study of domain theory and continuous lattices whenstudying canonical extensions:

• Order topologies. Order topologies are used heavily in domain theory, andthey occur very naturally when studying canonical extensions [39]. Weoffer the most extensive treatment of the topological properties of canonicalextensions to date, and we present two results in §2.1.3 which make clearthat the topologies on canonical extensions are both central, even defining,and natural.

• Filter and ideal completions. Many results about canonical extensions canbe understood by looking at an intermediate level of filters and ideals, ratherthan only at the canonical extensions themselves [44]. In fact, the filterand ideal completion functors play a central role in the theory of canonicalextensions of order-preserving maps – a fact which was foreshadowed in[41]. We revisit and expand upon the results of [34], showing how the filtercompletion and the ideal completion play an important role ‘under the hood’.

• Dcpo presentations. Directed complete partial orders are central in domaintheory. We already indicated that the canonical extension is intimatelyconnected with the filter completion and the ideal completion. In fact, wecan present the canonical extension of a lattice L as a dcpo generated bythe filter completion F L.

This chapter is organized as follows. In §2.1, we introduce canonical extensionsof lattices, both classically and using a new topological characterization in §2.1.3.In §2.2, we develop the theory of canonical extensions of order-preserving mapswith an emphasis on the role of the filter and ideal completion. Finally, in §2.3we present an alternative characterization of canonical extensions using dcpopresentations. Furthermore, at the end of each section we provide a discussion ofthe contributions in that section and suggestions for further work.

2.1 Canonical extension via filters, ideals and

topology

In this section we want to introduce the canonical extension of a bounded lattice,which is a well-studied lattice completion, together with an improved topologicalperspective on this completion. The key to this topological perspective lies inunderstanding the role that the ideal and the filter completion play with respectto the canonical extension.

In §2.1.1, we introduce the classical definition of canonical extensions of lattices.In §2.1.2, we will introduce several important topologies on partial orders and

2.1. Canonical extension via filters, ideals and topology 11

lattice completions. In §2.1.3, we will present the two main new results of thissection, which characterize canonical extensions of lattices topologically, and whichidentify the characterizing topologies of canonical extensions. Finally, in §2.1.4,we present assorted additional properties of canonical extensions of lattices thatwill be of use later on.

2.1.1 The canonical extension of a lattice

In this subsection, we will introduce the canonical extension Lδ of a lattice L. Wewill define a concrete construction on a given lattice, using filters and ideals, andwe will then state a uniqueness result which tells us that any completion of Lsatisfying certain abstract order-theoretic properties is in fact isomorphic to Lδ.From that point on, we will no longer concern ourselves with the actual concreteconstruction of Lδ; rather, we will show how one can understand Lδ through itsabstract characterization.

Overlapping sets, filters and ideals

Before we go ahead and introduce the canonical extension, we would like tointroduce a very elegant concept from constructive mathematics. We will oftenwant to talk about sets U, V which have a non-empty intersection, i.e. U ∩ V 6= ∅.We would like to think of this relation on sets as a positive property.

2.1.1. Definition. Given two sets U, V ⊆ X, we write U G V (U and V overlap)if U ∩ V 6= ∅.

What makes the overlapping relation interesting is that it interacts nicely withseveral operations and relations on sets.

2.1.2. Lemma. Let X be a set and let U, V ⊆ X such that U G V .

1. If U ′, V ′ ⊆ X such that U ⊆ U ′ and V ⊆ V ′, then also U ′ G V ′.

2. If f : X → Y is a function to another set Y , then also f [U ] G f [V ].

Consequently, if f : P→ Q is an order-preserving map and if F ∈ F P, I ∈ I Psuch that F G I, then also F f(F ) G I f(I).

Proof Parts (1) and (2) are elementary. For the last part it suffices to recall thatF f(F ) := ↑ f [F ] and I f [F ] := ↓ f [F ].

The following lemma provides a further indication that the overlapping relationalso interacts in a nice way with filters and ideals. Recall from §A.5.1 thatgiven a poset P, I P := 〈IdlP,⊆〉 is the ideal completion of P, and dually thatF P := 〈FiltP,⊇〉 is the filter completion of P.


2.1.3. Lemma. Let P be a poset. If F ∈ F P and I ∈ I P such that F G I, then↓(F ∩ I) = I and ↑(F ∩ I) = F .

Proof We only show the first case, since the other follows by order duality. Takey ∈ I, we will show that there exists z ∈ F ∩ I such that y ≤ z. Since F G I,there exists x ∈ F ∩ I. Because x, y ∈ I, there exists z ∈ I such that x ≤ z andy ≤ z. Because x ∈ F and F is an upper set, z ∈ F . But then y ≤ z ∈ F ∩ I, sothat I ⊆ ↓(F ∩ I). The other inclusion follows immediately from the fact thatF ∩ I ⊆ I and I is a lower set.

Existence and uniqueness of the canonical extension

We will now present the canonical extension first as a concrete construction onlattices, and later as a completion of lattices which is unique up to isomorphism.The particular concrete construction we have chosen, which seems to go back to[44], is not the only possible one. In light of the uniqueness theorem however, theconcrete construction of the canonical extension we choose now is not particularlyimportant. Our construction will be a two-stage construction on a given latticeL. The first stage in the construction consists of creating a pre-order. The orderrelation we use goes back to [44].

2.1.4. Definition. Let L be a lattice. We define a structure IntL := 〈F L ]I L,v〉, where for all F, F ′ ∈ F L and I, I ′ ∈ I L,

F v I if F G I;F v F ′ if F ⊇ F ′;I v I ′ if I ⊆ I ′;I v F if I × F ⊆ ≤L,

i.e. I v F iff for all a ∈ I and for all b ∈ F , we have a ≤ b.

The following fact is well-known, cf. [44, p. 11].

2.1.5. Lemma. Let L be a lattice. Then IntL is a pre-order.

Proof It suffices to show that v is transitive. For this, we need to make a casedistinction. Let F, F ′ ∈ F L and I, I ′ ∈ I L.

F v F ′ v I ⇒ F v I by Lemma 2.1.2;

F v I v I ′ ⇒ F v I ′ idem;

I v I ′ v F ⇒ I v F easy,


since if I ⊆ I ′ and I ′ × F ⊆ ≤L then I × F ⊆ I ′ × F ⊆ ≤L;

I v F v F ′ ⇒ I v F ′ idem;

I v F v I ′ ⇒ I v I ′ see below;

F v I v F ′ ⇒ F v F ′ see below.

The last two cases have essentially the same proof; we only discuss the latter.If F v I v F ′, i.e. if F G I and I × F ′ ⊆≤L, then we need to show that alsoF v F ′, i.e. that F ⊇ F ′. Let a ∈ F ′. Since F G I, there exists b ∈ F ∩ I. SinceI × F ′ ⊆≤L, we have b ≤ a. But then a ∈ F ; it follows that F ⊇ F ′.

Note that in other places where IntL is introduced [44, 32, 41], one also takes aquotient of the pre-order to make it into a partial order. We do not bother withthis because the pre-order is flattened into a partial order in the second stage ofconstructing the canonical extension anyway. This second stage consists of takingthe MacNeille completion (see §A.5.2) of IntL.

2.1.6. Definition. We define the canonical extension of L to be Lδ := IntL ,with i : IntL→ Lδ the embedding of the MacNeille completion. We map L to Lδby setting eL : a 7→ i(↓ a).

We now have a way to constuct a complete lattice Lδ, given a lattice L, and afunction eL : L → Lδ. We see below that eL is in fact a lattice completion, andthat we can characterize it up to isomorphism of completions.

2.1.7. Definition. Given a (bounded) lattice L and a complete lattice C, wecall a lattice embedding e : L→ C a completion of L. We say two completions ofe : L→ C and e′ : L→ C′ are isomorphic if there exists an isomorphism h : C→ C′such that he = e′.

C h // C′

L

e

OO

e′

>>~~~~~~~~

Before we state the basic uniqueness result concerning canonical extensions, wewould like to point out that any completion e : L→ C induces two auxiliary mapseF : F L→ C and eI : I L→ C, if we exploit the fact that a complete lattice Cis simultaneously a dcpo and a co-dcpo.

2.1.8. Definition. Given a lattice completion e : L → C, we define eI : I L→ Cby I 7→

∨↑ e[I]. It is easy to see that eI : I L→ C is the unique Scott-continousextension of e : L→ C, the existence of which is stipulated by Fact A.5.3.

I LeI

L

↓==||||||||

e// C


Dually, we define eF : F L→ C by F 7→∧↓ e[F ]. We emphasize that eI(↓x) =

eF(↑x) = e(x) for all x ∈ L. We refer to eF(F ) | F ∈ I L and eI(I) | I ∈ I Las the filter and ideal elements of C, respectively.1

Filter and ideal elements play a crucial role in defining and understanding thecanonical extension.

2.1.9. Fact ([41], Proposition 3.6). Let L be a lattice, then the map eL : L→Lδ defined above is a lattice embedding. Moreover, if e : L→ C is a completion ofL such that

1. for all x ∈ C,

x =∨

eF(F ) | eF(F ) ≤ x, F ∈ F L =∧eI(I) | eI(I) ≥ x, I ∈ I L

;

2. for all F ∈ F L, I ∈ I L, if eF(F ) ≤ eI(I) then F G I;

then there exists a (unique) isomorphism of completions h : C→ Lδ, i.e. such thathe = eL.

We will refer to a lattice completion e : L→ C satisfying the conditions above asa canonical extension of L. The first condition of Fact 2.1.9, which is traditionallycalled density, states that given a canonical extension e : L→ C,

• the filter elements of C are join-dense, and

• the ideal alements of C are meet-dense.

The second condition (traditionally known as compactness) could also have beenstated as:

• for all F ∈ F L, I ∈ I L, eF(F ) ≤ eI(I) iff F G I,

since F G I implies that there is a ∈ F ∩ I, so that∧e[F ] ≤ e(a) ≤

∨e[I]. From

here on, we will simply refer to the density and compactness properties of thecanonical extension instead of explicitly referring to Fact 2.1.9.

We conclude this subsection with a class of examples of canonical extensionswhich is both important and trivial. Recall that a poset P is said to satisfy theascending chain condition (ACC) if for every countable chain x0 ≤ x1 ≤ x2 ≤· · · ≤ xi ≤ · · · , there exists a n ∈ N such that for all k ∈ N, xk ≤ xn.

2.1.10. Fact. Let L be a (bounded) lattice. The identity embedding idL : L→ Lis a canonical extension of L iff L satisfies both the ascending chain condition(ACC) and the descending chain condition (DCC).

1This is an abuse of language, since strictly speaking they are the filter and ideal elements ofe : L→ C.


2.1.11. Example. Examples of lattices satisfying ACC and DCC include finitelattices, and lattices such as M∞, where

M∞ := 〈0, 1 ∪ an | n ∈ N,≤〉,

and x ≤ y iff x = 0 or y = 1, see Figure 2.1.

a0 a1 a2 an

0

1

Figure 2.1: The lattice M∞ is a fixed point of the canonical extension

2.1.12. Convention. If L is a finite lattice, then we define Lδ := L in light ofFact 2.1.10.

2.1.2 Topologies on posets and completions

We will now introduce two families of topologies which are defined on posets,and one which is defined on completions. The Scott topologies and the intervaltopologies are defined on any poset. The δ topologies are defined on any latticecompletion. All three families come in three kinds: one where every open set isan upper set, one where every open set is a lower set, and the join of these upperand lower topologies, where every basic open is an intersection of an open upperset and an open lower set.

2.1.13. Definition. Let P = 〈P,≤〉 be a poset.

• By ι↑(P) := 〈P \ ↓x | x ∈ P〉 we denote the upper interval topology of P,and ι↓(P) := ι↑(Pop).

• By σ↑(P) we denote the Scott topology on P: U ⊆ P is σ↑-open if U is anupper set which is inaccessible by directed joins, or equivalently if P \ Uis a lower set closed under all existing directed joins. By σ↓(P) we denoteσ↑(Pop).


Let e : L→ C be a lattice completion. We define two topologies on C:

• δ↑(C) :=⟨↑ eF(F ) | F ∈ F L

⟩and δ↓(C) :=

⟨↓ eI(I) | I ∈ I L

⟩.

Recall from §A.7 that if τ and τ ′ are topologies on a set X, then τ ∨ τ ′ :=⟨U ∩ V | U ∈ τ, V ∈ τ ′

⟩is the least topology on X containing τ and τ ′. We

define σ(P) := σ↑(P) ∨ σ↓(P) (the bi-Scott topology) and δ(C) := δ↑(C) ∨ δ↓(C).Below, if e.g. f : C→M is some map, we will say that f is (δ↑, σ↑)-continuous

if f : 〈C, δ↑(C)〉 → 〈M,σ↑(M)〉 is a continuous function.

2.1.14. Remark. Observe that given a completion e : L → C, the filter and idealelements of C are often referred to as the ‘closed’ and ‘open’ elements of C in thecanonical extension literature (cf. [34, Lemma 3.3]). This makes our definitions ofthe δ↑ and δ↓ topologies equivalent with those in [39], where they are called σ↑

and σ↓ respectively.Although we will not use it, it is worth noting that ι := ι↑ ∨ ι↓ is the usual

interval topology. For example, ι(R) is the topology generated by

z ∈ R | x < z < y | x, y ∈ R ,

i.e. the usual topology on the real line.

2.1.15. Lemma. Let e : L→ C be a lattice completion.

1. The following set is a base for δ↑(C):

↑∨F∈Se

F(F ) | S ⊆ F L finite.

2. The following set is a base for δ↓(C):

↓∧I∈T e

I(I) | T ⊆ I L finite.

3. The following set is a base for δ(C):

↑∨F∈Se

F(F ) ∩ ↓∧I∈T e

I(I) | S ⊆ F L, T ⊆ I L finite.

Proof It suffices to show that (1) holds. First, observe that if S ⊆ F L, thensince C is complete, it follows by order theory that⋂

F∈S ↑ eF(F ) = ↑

∨F∈Se

F(F ). (2.1)

Next, observe that since ↑ eF(F ) | F ∈ F L is a subbase for δ↑(C) by definition,it follows by general topology that

⋂F∈S ↑ e

F(F ) | S ⊆ F L finite

is a base for δ↑(C). It now follows by (2.1) that (1) holds.


We are considering topologies defined on ordered sets, and the order playsan important role in defining these topologies. Below we will establish someelementary facts about the interaction between our topologies and the order.

2.1.16. Lemma. Let P be a poset. If O ⊆ P is σ-open, then P \O is closed underall existing directed joins and codirected meets.

Proof We only show the case of directed joins. Towards a contradiction, letS ⊆ P \ O be a directed set such that

∨S ∈ O. Then by definition of σ, there

must exist a σ↑-open U ⊆ P and a σ↓-open V ⊆ P such that∨S ∈ U ∩ V ⊆ O.

Since U is Scott-open and∨S ∈ U , there is some x ∈ S ∩ U . Since V is a lower

set, we also get x ∈ V , which is a contradiction.

2.1.17. Lemma. Let P be a poset and let e : L→ C be a lattice completion.

1. ι↑(P) ⊆ σ↑(P) and ι↓(P) ⊆ σ↓(P);

2. U ∈ σ(P) | U is an upper set = σ↑(P);

3. U ∈ δ(C) | U is an upper set = δ↑(C).

Consequently, an order-preserving map f : P→ Q is (σ, σ)-continuous iff it is both(σ↑, σ↑)-continuous and (σ↓, σ↓)-continuous.

Proof (1). This is easy to see: any subbasic ι↑-open ↓x is also σ↑-open, sinceobviously ↓x is a lower set closed under directed joins.

(2). Suppose that U is a σ-open set such that U is an upper set. Then allwe have to do to show that U is σ↑-open, is to show that P \ U is closed underdirected joins. But this follows immediately from Lemma 2.1.16.

(3). Suppose that U ⊆ C is a δ-open upper set. To show that U is δ↑-open,it suffices to show that for every x ∈ U , there exists a δ↑-open U ′ ⊆ U such thatx ∈ U ′. Take x ∈ U . Since U is δ-open, by Lemma 2.1.15(3), there exist finitesets S ⊆ F L and T ⊆ I L such that

x ∈ ↑∨F∈Se

F(F ) ∩ ↓∧I∈T e

I(I) ⊆ U.

Then∨F∈Se

F(F ) ∈ U , so since U is an upper set,

U ′ := ↑∨F∈Se

F(F ) ⊆ U.

By Lemma 2.1.15(1), U ′ is δ↑-open; since x ∈ U ′ ⊆ U was arbitrary it follows thatU is δ-open.

For the last claim of the lemma, suppose that f : P→ Q is order preserving.If f is both (σ↑, σ↑) and (σ↓, σ↓)-continuous, then it follows by general topology(Lemma A.7.3) that f is (σ↑ ∨ σ↓, σ↑ ∨ σ↓)-continuous, so since σ := σ↑ ∨ σ↓, fis (σ, σ)-continuous. Conversely, if f is (σ, σ)-continuous and U ⊆ Q is an upperset, then f−1(U) is also an upper set since f is order-preserving. Now by part(2) above, f−1(U) is σ↑-open; since U ⊆ Q was arbitrary, it follows that f is(σ↑, σ↑)-continuous. The argument for (σ↓, σ↓)-continuity is analogous.


When forming, say, the topology σ = σ↑ ∨ σ↓ on a poset P, we are creatingmany new open sets. A priori, it is possible that σ contains new open upper setswhich are not σ↑-open. The lemma above tells us that in the case of the σ and δtopologies, this does not happen.

Recall that the σ and ι-topologies were defined using order duality: ι↓(P) :=ι↑(Pop). The following lemma states that we could have done the same with the δtopologies.

2.1.18. Lemma. Let e : L → C be a lattice completion. Then the followingtopologies on C coincide: δ↑(L) = δ↓(Lop). Consequently, δ(L) = δ(Lop).

Proof This follows easily from Fact A.5.2.

Of course, the above lemma also holds, by definition, for the Scott topology.At this point we come to a property which the Scott topology notoriously lacks.

2.1.19. Lemma. Let e1 : L1 → C1 and e2 : L2 → C2 be two lattice completions.Then the following topologies on C1×C2 coincide: δ↑(C1)× δ↑(C2) = δ↑(C1×C2)and δ↓(C1)× δ↓(C2) = δ↓(C1 × C2).

Proof We will show that δ↑(C1)× δ↑(C2) = δ↑(C1 × C2). Recall that

δ↑(C1 × C2) = 〈↑C1×C2(e1 × e2)F(F ) | F ∈ F(L1 × L2)〉

= 〈↑C1×C2(eF1 (F1), eF2 (F2)) | (F1, F2) ∈ F L1 ×F L2〉,

where the last equality follows from fact that F commutes with products (FactA.5.4). Moreover, if (F1, F2) ∈ F L1 × F L2 so that (eF1 (F1), e

F2 (F2)) ∈ C1 × C2,

then again by Fact A.5.4 (applied to C1 × C2), we see that

↑C1×C2(eF1 (F1), eF2 (F2)) = ↑C1

(eF1 (F1)

)× ↑C2

(eF2 (F2)

).

But we know from general topology that

δ↑(C1)× δ↑(C2) = 〈↑C1

(eF1 (F1)

)× ↑C2

(eF2 (F2)

)| F1 ∈ F L1, F2 ∈ F L2〉,

so it follows that δ↑(C1)× δ↑(C2) = δ↑(C1 × C2).

The above lemma is not true of the Scott topology: there exist lattices L, Msuch that σ↑(L×M) 6= σ↑(L)× σ↑(M) [45, Thm. II-4.11].

2.1.3 Characterizing the canonical extension via the δ-topology

In this subsectoin, we present the two main new results of this section. Firstly, weprove a new, topological characterization theorem for the canonical extension. Sec-ondly, we show how the δ-toplogies can be given a natural description as subspacetopologies with respect to the Scott and co-Scott topology on superstructures ofLδ.


The interaction between the δ-topologies and the maps eF and eI

We start with a lemma which tells us that the compactness property of canonicalextensions is in fact a topological property.

2.1.20. Lemma. Let e : L → C be a completion of L. Then the following areequivalent:

1. for all F ∈ F L and I ∈ I L, eF(F ) ≤ eI(I) iff F G I.

2. eI : I L→ C is (σ↑, δ↑)-continuous and eF : F L→ C is (σ↓, δ↓)-continuous.

Proof Assume (1) holds. Let ↑ eF(F ) be a subbasic open set of δ↑ (for someF ∈ F L). We will show that U := (eI)−1

(↑ eF(F )

)is Scott-open in I L. Since

eI is order-preserving, we see that U must be an upper set. Now let S ⊆ I L bedirected; then

∨S =

⋃S. If

∨S ∈ U , then

eI(∨S) = eI(

⋃S) ∈ ↑ eF(F ),

i.e. eI(⋃S) ≥ eF(F ). It follows by (1) that there is a ∈ F ∩

⋃S, i.e. there

is an I ∈ S such that a ∈ F ∩ I. But then F G I, so by (1), it follows thateF(F ) ≤ eI(I), so that I ∈ (eI)−1

(↑ eF(F )

)= U . Since S was arbitrary it follows

that U is Scott-open. The proof of the statement for eF is the order dual of theabove; it follows that (2) holds.

Conversely, assume that (2) holds and let F ∈ F L, I ∈ I L. If F G I thenthere is a ∈ F ∩ I, so we see that

eF(F ) =∧e[F ] ≤ e(a) ≤

∨e[I] = eI(I).

Now suppose that eF(F ) ≤ eI(I), so that I ∈ U := (eI)−1(↑ eF(F )). By (2), U isScott-open. Now since I =

∨a∈I ↓ a is a directed join, it follows that there is some

a ∈ I such that ↓ a ∈ U , i.e. eI(↓ a) ∈ ↑ eF(F ). Equivalently, eF(F ) ∈ ↓ eI(↓ a),so by an argument analogous to that above, there is some b ∈ F such thateF(↑ b) ≤ eI(↓ a). But now

e(b) = eF(↑ b) ≤ eI(↓ a) = e(a),

so that b ≤ a. Since a ∈ I and I is a lower set, we also get b ∈ I, so that F G I.It follows that (1) holds.

The above lemma effectively translates one of the defining properties (compactness)of the canonical extension into a topological property of lattice completions. Thefollowing lemmas establish some topological properties of eF and eI which willbe of use later. First, we show that eI : I L → C is a Scott-continuous latticehomomorphism under certain topological assumptions.


2.1.21. Lemma. Let L be a bounded lattice and let e : L → C be a completion ofL.

1. eI : I L→ C preserves all joins (including 0), and the top element 1.

2. If eI : I L → C is (σ↑, δ↑)-continuous and δ↑ is T0, then eI : I L → Cpreserves finite meets.

Proof (1). Preservation of 0 and 1 follows from the fact that e : L → C is abounded lattice embedding; if we look at e.g. the top element of I L, i.e. ↓ 1, theneI(↓ 1) =

∨e[↓ 1] = e(1) = 1. The argument for 0 is identical. Now for binary joins,

recall that the join of two ideals I1, I2 ∈ I L is I1∨I2 := ↓a1∨a2 | a1 ∈ I1, a2 ∈ I2.Now

eI(I1 ∨ I2) =∨e[I1 ∨ I2] =

∨e(a1 ∨ a2) | a1 ∈ I1, a2 ∈ I2 =∨

e(a1) ∨ e(a2) | a1 ∈ I1, a2 ∈ I2 =∨e[I1] ∨

∨e[I2] = eI(I1) ∨ eI(I2),

where the penultimate equality follows from the fact that I1 and I2 are directed.Since eI is Scott-continuous by Fact A.5.3, it follows that eI preserves all joins.

(2). We will only consider binary meets. Assume that δ↑ is T0 and let I1, I2 ∈I L. Since eI is order-preserving, we only need to show that eI(I1) ∧ eI(I2) ≤eI(I1∩I2). Suppose not, then since δ↑ is T0 there must be some δ↑-open U such thateI(I1)∧eI(I2) ∈ U and eI(I1∩I2) /∈ U . We see that also eI(I1) ∈ U and eI(I2) ∈ U ;moreover, without loss of generality we may assume U =

⋂1≤i≤n ↑ eF(Fn) for

some finite set F1, . . . , Fn ⊆ F L. Since eI : I L → C is (σ↑, δ↑)-continuous,(eI)−1(U) is Scott-open. Now since I1 =

∨a∈I1 ↓ a ∈ (eI)−1(U) is a directed

join, there must be some a1 ∈ I1 such that eI(↓ a1) = e(a1) ∈ U , and similarlythere must be some a2 ∈ I2 such that e(a2) ∈ U . Since U =

⋂1≤i≤n ↑ eF(Fn),

it follows that for all 1 ≤ i ≤ n, we have that e(a1), e(a2) ∈ ↑ eF(Fi), so alsoe(a1) ∧ e(a2) = e(a1 ∧ a2) ∈ ↑ eF(Fi). Since i was arbitrary, it follows thate(a1 ∧ a2) ∈ U . Since we also have that a1 ∧ a2 ∈ I1 ∩ I2, it follows that

eI(I1 ∩ I2) =∨e[I1 ∩ I2] ≥ e(a1 ∧ a2) ∈ U,

which is a contradiction since we assumed U is an upper set not containingeI(I1 ∩ I2). It follows that eI(I1 ∩ I2) = eI(I1) ∧ eI(I2).

Next, we show that under the assumptions of Lemma 2.1.21, the δ↑-topologyon a completion e : L→ C has a base of principal lower sets.

2.1.22. Corollary. Under the assumptions of Lemma 2.1.21, ↓ eI(I) | I ∈I L is not only a subbase for δ↓, but in fact a base.

Proof We will show that ↓ eI(I) | I ∈ I L is closed under finite intersections.Let I1, I2 ∈ I L, then

↓ eI(I1) ∩ ↓ eI(I2) = ↓(eI(I1) ∧ eI(I2)) = ↓ eI(I1 ∩ I2).


The δ-topology determines the canonical extension

Now that we have proved the required technical results about the δ-topologies, wecan present the first main result of this section: we show how the δ-topologies,which arise naturally on canonical extensions, in fact characterize it.

2.1.23. Theorem. Let L be a bounded lattice and let e : L → C be a completionof L. Then e : L → C is a canonical extension of L iff

1. eI : I L→ C is (σ↑, δ↑)-continuous and eF : F L→ C is (σ↓, δ↓)-continuous,

2. δ↑ and δ↓ are both T0.

Proof First assume that e : L→ C is a canonical extension of L. It follows fromLemma 2.1.20 that (1) holds. Moreover, if x, y ∈ C and x y, then there mustbe some F ∈ F L and I ∈ I L such that eF(F ) ≤ x and eF(F ) y. It followsthat x ∈ ↑ eF(F ) and y /∈ ↑ eF(F ), so δ↑ is T0. The proof for δ↓ is analogous; itfollows that (2) holds.

Conversely, assume that (1) and (2) hold. It follows from Lemma 2.1.20 thatcondition (2) of Fact 2.1.9 holds. Moreover, if x, y ∈ C and x y, then since δ↑ isT0, there must exist some finite S ⊆ F L such that

x ∈⋃F∈Se

F(F ) 63 y.

It follows that there must be some F ∈ S such that y /∈ ↑ eF(F ); now we seethat eF(F ) ≤ x and eF(F ) y. An analogous argument shows that since δ↓ isT0, there must be some I ∈ I L such that y ≤ eI(I) and x eI(I). Since x, yCwere arbitrary, it follows that (1) of Fact 2.1.9 holds, so e : L→ C is a canonicalextension of L.

Explaining the δ-topology

We now present the second main result of this section, which sheds a different lighton the definition of the δ-topology. In particular, we will focus on the δ↑-topology.We will show that there is a natural way to embed Lδ in I F L, and that theδ↑-topology on Lδ is simply the Scott topology on I F L, restricted to Lδ.

So how do we embed Lδ in I F L? The key insight is that since eF : F L→ Lδpreserves all finite joins (by Lemma 2.1.24), the map I eF : I F L→ I Lδ has aleft adjoint, namely (eF)−1 : I Lδ → I F L.

2.1.24. Lemma. Let L,M be lattices and let f : L → M be a map preserving ∨and 0.

1. The inverse image function f−1 maps ideals of M to ideals of L;


2. I f a f−1, i.e. for all I ∈ I L, J ∈ IM, we have

I f(I) ⊆ J iff I ⊆ f−1(J);

Proof (1). If J ∈ IM, then f−1(J) is a lower set since f is order-preserving.Moreover, if a, b ∈ f−1(J), then f(a), f(b) ∈ J , so f(a) ∨ f(b) = f(a ∨ b) ∈ J , sothat a ∨ b ∈ f−1(J). Finally, since 0 = f(0) ∈ J , it follows that 0 ∈ f−1(J), sothat f−1(J) is non-empty.

(2). If I f(I) = ↓ f [I] ⊆ J , then for every a ∈ I, f(a) ∈ J , i.e. a ∈ f−1(J), soI f(I) ⊆ J implies I ⊆ f−1(J). Conversely, if I ⊆ f−1(J), then for every a ∈ I,f(a) ∈ J , i.e. f [I] ⊆ J . Since J is a lower set, we also get I f(I) = ↓ f [I] ⊆ J .

We now define g : Lδ → I F L as g := (eF)−1 ↓Lδ . This map g will be theembedding that shows that Lδ is isomorphic to a subposet of I F L.

2.1.25. Theorem. Let e : L→ Lδ be a canonical extension.

1. The following diagram commutes:

I F LI eF //

I Lδ(eF )−1

oo

F L

↓F L

OO

eF// Lδ↓Lδ

OO

2. The composite g := (eF)−1 ↓Lδ is a (δ↑, σ↑)-continuous homeomorphicembedding.

Proof (1). We need to show that

I eF ↓F L = ↓Lδ eF (2.2)

and that(eF)−1 ↓Lδ eF = ↓F L. (2.3)

The validity of (2.2) follows from the fact that ↓ is a natural transformation byFact A.5.3(1). To see why (2.3) holds, first observe that since eF : F L → Lδpreserves all finite joins by Lemma 2.1.21, it follows by Lemma 2.1.24(2) that(eF)−1 : I Lδ → I F L is right adjoint to I eF . Since I eF is an order-embeddingby Fact A.5.3(5), it follows by Fact A.3.3(2) that

(eF)−1 I eF = idI F L . (2.4)

Now we see that

(eF)−1 ↓Lδ eF = (eF)−1 I eF ↓F L by (2.2),

= ↓F L by (2.4).


(2). We now define g := (eF)−1 ↓Lδ ; observe that

g(x) = (eF)−1(↓Lδ x) = F ∈ F L | eF(F ) ≤ x, (2.5)

which is an ideal of F L. This map will be the embedding of Lδ into I F L. Con-versely, there is a natural map from I F L to Lδ: since eF : F L→ Lδ is an orderpreserving map from F L to a dcpo Lδ, the universal property of ideal completiontells us that there exists a unique Scott-continuous map (eF)I : I F L→ Lδ suchthat (eF)I ↓F L = eF ; namely

(eF)I : I 7→∨eF [I] =

∨eF(F ) | F ∈ I, (2.6)

where I ∈ I F L.

I F L(eF )I

F L

eF//

↓F L::vvvvvvvvvLδ

Now observe that (eF)I g = idLδ : take x ∈ Lδ, then

(eF)I g(x)

=∨eF(F ) | F ∈ g(x) by (2.6),

=∨eF(F ) | eF(F ) ≤ x by (2.5),

= x by join-density of filter elements.

Now it is easy to see that g is an order embedding:

g(x) ≤ g(y)

⇒ (eF)I g(x) ≤ (eF)I g(y) since (eF)I is order-preserving,

⇒ x ≤ y since (eF)I g = idLδ .

We will now show that g : Lδ → I F L is a (δ↑, σ↑)-continuous homeomorphicembedding, meaning that for every δ↑-open U ⊆ Lδ there exists a σ↑-openU ′ ⊆ I F L such that U = g−1(U ′). It suffices to show this for the case that U isa basic open, i.e. for the case that U = ↑Lδ e

F(F ) for an arbitrary F ∈ F L. SinceF ∈ F L, we know that ↓F L F is a compact element of I F L, so that ↑I F L(↓F L F )is σ↑-open. Now

g−1 (↑I F L ↓F L F )

= g−1(↑I F L g eF(F )

)since g eF = ↓F L,

= y ∈ Lδ | g eF(x) ≤ g(y) by def. of g−1 and ↑,= y ∈ Lδ | eF(F ) ≤ y because g is an ord. emb.,

= ↑Lδ eF(F ).

It follows that g is a homeomorphic embedding.


The fact that g : Lδ → I F L is a (δ↑, σ↑)-homeomorphic embedding tells us twothings. Firstly, that Lδ is isomorphic to a subposet of I F L. Secondly, this tellsus that the δ↑-topology is precisely the topology that Lδ inherits as a subspace ofI F L, where the latter is endowed with the σ↑-topology. Naturally, this resultdualizes if we view Lδ as a subposet of F I L.

2.1.4 Basic properties of the canonical extension

In this section, we will introduce assorted basic properties of canonical extensionswhich will be of use later on. The first observations concern the interaction withproducts and the operation of taking the order dual of a lattice. After that, wewill prove certain topological and order-theoretical properties of the maps eI andeF and of the δ-topologies. In particular, we will see that canonical extensionssatisfy two distributive laws with respect to joins of filter elements and meetsof ideal elements. We conclude the subsection with a result about the internalstructure of the lattices that arise as canonical extensions.

We begin by showing that canonical extensions commute with finite productsand order duals of lattices.

2.1.26. Lemma. Let e1 : L1 → Lδ1 and e2 : L2 → Lδ2 be canonical extensions.

1. e1 × e2 : L1 × L2 → Lδ1 × Lδ2 is a canonical extension of L1 × L2.

2. eop1 : Lop1 → (Lδ1)op is a canonical extension of Lop1 .

Proof (1). We will verify that e1×e2 : L1×L2 → Lδ1×Lδ2 satisfies the topologicalconditions of Theorem 2.1.23. Since the product of two T0 spaces is again T0, itfollows from Lemma 2.1.19 that both δ↑(Lδ1 × Lδ2) and δ↓(Lδ1 × Lδ2) are T0.

Now since eIi : I Li → Lδi is (σ↑, δ↑)-continuous for i = 1, 2, it follows fromFact A.5.4, Lemma 2.1.19 and general topology that

eI1 × eI2 : I L1 × I L2 → Lδ1 × Lδ2 is (σ↑, δ↑)-continuous.

Let h : I(L1 × L2) → I L1 × I L2 be the order-isormorphism witnessing thatI(L1×L2) ' I L1×I L2; observe that h is (σ↑, σ↑)-continuous. Now consider thediagram in Figure 2.2. The upper left triangle commutes by Fact A.5.4, the upperright triangle commutes by the universal property of ↓L1×L2

: L1×L2 → I(L1×L2)(Fact A.5.3), and the lower triangle commutes by definition of (e1 × e2)

I . Nowsince (eI1 × eI2 ) h is (σ↑, δ↑)-continuous, so is (e1 × e2)I , which is what we neededto show. The argument showing that (e1×e2)F : F(L1×L2)→ Lδ1×Lδ2 is (σ↓, δ↓)-continuous is identical; it follows by Theorem 2.1.23 that e1×e2 : L1×L2 → Lδ1×Lδ2is a canonical extension of L1 × L2.

(2). This follows readily from the order duality between filters and ideals andLemma 2.1.18.


Figure 2.2: Finite products of canonical extensions

I L1 × I L2

eI1×eI2

>>>>>>>>>>>>>>>>>>>

I(L1 × L2)

(e1×e2)I &&NNNNNNNNNNN

h

OO

L1 × L2

↓L1×↓L2

@@ ↓L1×L2

88ppppppppppp

e1×e2// Lδ1 × Lδ2

The fact that canonical extension commutes with taking finite products is infact an instance of the more powerful result that canonical extensions commutewith Boolean products (see §A.6). An alternative reference for the following resultis [43, Theorem 5.1].

2.1.27. Fact ([34]). Let L be a lattice and let (px : L → Mx)x∈X be a Booleanproduct decomposition of L. Then Lδ ∼=

∏XMδ

x.

Parts (1) and (2) of the following lemma can be found in [41]. Part (3) however,which we will use very frequently, is new. In light of Theorem 2.1.25, part (3)below is perhaps not that surprising.

2.1.28. Lemma. Let e : L→ Lδ be the canonical extension of a bounded latticeL.

1. (a) eF : F L→ Lδ is an ∨,∧

-embedding;

(b) eI : I L→ L is an ∧,∨

-embedding;

2. for all x ∈ Lδ,

(a) eF(F ) | eF(F ) ≤ x is directed;

(b) eI(I) | x ≤ eI(I) is co-directed;

3. (a) σ↑(Lδ) ⊆ δ↑(Lδ);

(b) σ↓(Lδ) ⊆ δ↓(Lδ);

(c) σ(Lδ) ⊆ δ(Lδ);

4. e[L] is dense in 〈Lδ, δ(Lδ)〉 and in 〈Lδ, σ(Lδ)〉.

Proof (1). We will only prove part (a). It follows from (the order dual of) Lemma2.1.21 that eF is a ∨,

∧-homomorphism. To show that it is an embedding, we

will show that eF is order-reflecting. Assume that F + F ′; we will show thateF(F ) eF(F ′). By assumption, there exists a ∈ F ′ \ F , so that F ∩ ↓ a = ∅


and F ′ G ↓ a. It follows by the compactness property of canonical extension thateF(F ) eI(↓ a) and eF(F ′) ≤ eI(↓ a). Now by the meet-density of ideal elements,it follows that eF(F ) eF(F ′).

(2). We only show part (a). This follows from (1), since ↓x is an ideal.eF(F ) | eF(F ) ≤ x

(3). Suppose U ⊆ Lδ is Scott-open and that x ∈ U . Then x =∨eF(F ) |

eF(F ) ≤ x is a directed join by (1), so there must be some F such that eF(F ) ≤ xand eF(F ) ∈ U . It follows that x ∈ ↑ eF(F ) ⊆ U ; hence U is δ↑-open.

(4). Consider a non-empty basic open U of 〈Lδ, δ(Lδ)〉, i.e. there are F ∈ F L,I ∈ I L such that U = x ∈ Lδ | eF(F ) ≤ x ≤ eI(I). Since U 6= ∅, it must bethat eF(F ) ≤ eI(I), so that F G I. Take a ∈ F ∩ I, then

∧e[F ] ≤ e(a) ≤

∨e[I]

so that e(a) ∈ U . It follows that e[L] is dense in 〈Lδ, δ(Lδ)〉. Since σ ⊆ δ by (3),it is also the case that e[L] is dense in 〈Lδ, σ(Lδ)〉.

The distributive law below, which is similar to Lemma 3.2 of [34], is a verypowerful result. We will in fact use it to characterize the canonical extension as adcpo in §2.3.

2.1.29. Lemma. Let e : L→ Lδ be the canonical extension of a lattice L. Thenfor all S ⊆ F L and S ′ ⊆ I L,∨

eF(F ) | F ∈ S =∧eI(I) | ∀F ∈ S, F G I

and ∧eI(I) | I ∈ S ′ =

∨eF(F ) | ∀I ∈ S ′, F G I.

Proof We only prove the first statement. Since the ideal elements of Lδ aremeet-dense, we see that∨

eF(F ) | F ∈ S =∧eI(I) |

∨F∈Se

F(F ) ≤ eI(I).

Now observe that∨F∈Se

F(F ) ≤ eI(I) iff for all F ∈ S, eF(F ) ≤ eI(I) iff for allF ∈ S, F G I. The (first) statement of the lemma follows.

We conclude this subsection with a well-known result from the canonicalextension literature, which plays an important role in the duality theory forcanonical extensions. Let L be a complete lattice. Recall that an element p ∈ Lis called completely join-irreducible if for all S ⊆ L such that p =

∨S, there

exists a ∈ S such that p = a. We denote the set of completely join-irreducibleelements of L by J∞(L). Completely meet-irreducible elements are defined dually;we denote them by M∞(L). The following fact, which is related to Stone duality,requires the Axiom of Choice.

2.1.30. Fact ([34], Lemma 3.4). Let L be a lattice. Then Lδ is join-generatedby J∞(Lδ) and meet-generated by M∞(Lδ).

2.2. Canonical extensions of maps I: order-preserving maps 27

2.1.5 Conclusions and further work

This section served two purposes: to make the reader familiar with the classicaldefinition and well-known properties of the canonical extension, and to add anumber of fundamental topological results on canonical extensions. The work ofGehrke and Jonsson [39] has shown that the topological perspective on canonicalextensions is worthwile, however the basic topological properties of canonicalextensions in the general setting of (not necessarily distributive) lattices have notpreviously been studied. One fundamental difference between the distributive andthe non-distributive settings is that the Scott topology and the topology generatedby principal up-sets of completely join irreducibles no longer coincide.

All the results in §2.1.1 are known from the work of Gehrke and Harding [34],save perhaps the small technical results concerning the overlap relation. It shouldbe noted however that the emphasis on the auxiliary maps eF : F L → Lδ andeI : I L→ Lδ is a departure from the view on canonical extensions furthered in[34, 39].

In §2.1.3 we presented results which were first reported at TACL 2009 inAmsterdam. Both the topological characterization theorem (Theorem 2.1.23)and the result which casts the δ-topologies as subspace topologies (Theorem2.1.25) were previously unknown. The topological characterization theorem wasinspired by the work of Theunissen and Venema [86] on MacNeille completions oflattice-based algebras.

Further work

• It would be very interesting to see if Fact 2.1.27, which deals with canonicalextensions of Boolean products of lattices, can be given a new proof reducingit to a statement about the ideal completion and filter completion of lattices.

• It would also be interesting to see if there is a version of the topologicalcharacterization theorem (Theorem 2.1.23) which does not rely on eF andeI , while still giving a topological characterization of Lδ.

• An alternative approach to canonical extensions using filters and ideals hasbeen developed by Gehrke, Jansana & Palmigiano [37], using logical filtersrather than order filters. It would be interesting to see if their approach alsoadmits a topological characterization.

2.2 Canonical extensions of maps I: order-pre-

serving maps

In the previous section, we have studied the canonical extension as a constructionon lattices to some length. We will now turn to the subject of canonical extensions


of maps f : L → M between lattices, which is probably an even richer subject.In this dissertation, we will consider two approaches to obtaining a canonicalextension of a map f : L→M between lattices.

• The first approach is to assume that f is order-preserving, so that we mayfirst extend f : L → FM to maps F f : F L → M and I f : I L → IMacting on filters and ideals, respectively, and then work from there. This isthe approach we will take in the current section.

• The second approach is to assume that Mδ has nice topological properties, inwhich case we can develop a substantial part of the basic theory of extensionsof maps without making any assumptions about the map f : L→ M. Wewill pursue this approach in §3.2.

It is an interesting open question whether the two approaches above form adichotomy of some sort. We will return to this question in Remark 3.2.22.

In §2.2.1, we will first give the basic definition of the lower (fO) and theupper (fM) canonical extension of an order-preserving map f : L → M, and wewill characterize these extensions as a largest and smallest continuous extensionof f to a map Lδ → Mδ, respectively (Theorem 2.2.4). We will then show in§2.2.2 that if we assume that f preserves joins (or dually, meets), even in onlyone coordinate, then this vastly improves the behaviour of fO and fM (Theorem2.2.18). Finally in §2.2.3 we will put this good behaviour to use by showing thatcanonical extensions of lattice homomorphisms are particularly well-behaved, sowell in fact that canonical extension is a functor on the category of lattices andlattice homomorphisms (Theorem 2.2.24). All along, we will see that almost everyresult we prove about fO and fM reduces to statements about filters and ideals.

2.2.1 The lower and upper extensions of an order-preservingmap

In this subsection we will discuss the two canonical [34] ways to extend an order-preserving map f : L → M to a map f ′ : Lδ → Mδ, namely the lower and theupper canonical extension. We will then prove some of the basic facts that holdtrue for any order-preserving map. We conclude the subsection with a topologicalcharacterization theorem, which tells us that fO and fM can be seen as the largestand smallest continuous extension of f , respectively.

Consider the following diagram, where eL : L→ Lδ and eM : M→Mδ are thecanonical extensions of L and M, respectively:

L↑L //

f

F LeFL //

F f

Lδ

?

M ↑M// FM

eFM

//Mδ


(Observe that we factor eL : L→ Lδ as e(a) = eFL (↑ a).) We would like to find afunction that can take the place of the ‘?’ in the diagram. To do this, we use thefact that any x ∈ Lδ is approximated from below by Sx := F ∈ F L | eFL (F ) ≤ x.Now each F ∈ Sx can be mapped into Lδ via the assignment F 7→ eFM F f(F ).

2.2.1. Definition. Let f : L→M be an order-preserving map between lattices.Then we define fO : Lδ →Mδ, the lower extension of f , as follows:

fO : x 7→∨

eFM F f(F ) | eFL (F ) ≤ x.

Dually, we define fM : Lδ →Mδ, the upper extension of f , as follows:

fM : x 7→∧

eIM I f(I) | x ≤ eIL(I).

2.2.2. Remark. There is a slight discrepancy between our definition of fO andfM when we compare it with the working definition found in e.g. [34, Lemma 4.3].Using our notation, the working definition of [34] amounts to

fO : x 7→∨∧

eM f [F ] | eFL (F ) ≤ x.

The difference is that we use F f(F ) rather than f [F ]; however this difference isinconsequential. If F ∈ F L, then F f(F ) = ↑ f [F ] ⊇ f [F ], so

eFM (F f(F )) =∧eM[F f(F )]

]by def. of eFM,

≤∧eM f [F ] since F f(F ) ⊇ f [F ].

Conversely, since eM is order-preserving, we know by Fact A.3.1 that for all U ⊆M,↑ eM[U ] ⊇ eM[↑U ]. Now we see that∧

eM f [F ] =∧↑ eM f [F ] by order theory,

≤∧eM[↑ f [F ]

]since ↑ eM f [F ] ⊇ eM

[↑ f [F ]

].

In the following lemma we see that fO and fM mingle well with the auxilliarymaps induced by eL : L→ Lδ and eM : M→Mδ, or alternatively, that fO and fM

behave well on filter and ideal elements.

2.2.3. Lemma ([34]). Let f : L → M be order-preserving. Then fO : Lδ → Mδ

and fM : Lδ →Mδ are order-preserving maps, which satisfy the following additionalproperties:

1. fO eFL = eFM F f ;

2. fM eIL = eIM I f ;


3. fO ≤ fM;

4. fO eIL = fM eIL;

5. fM eFL = fO eFL .

Proof It is easy to see why fO and fM are order-preserving. Take x, y ∈ Lδ suchthat x ≤ y and consider fO. Then

F ∈ F L | eFL (F ) ≤ x ⊆ F ∈ F L | eFL (F ) ≤ y,

so also∨eFM F f(F ) ∈ F L | eFL (F ) ≤ x ≤

∨eFM F f(F ) ∈ F L | eFL (F ) ≤ y,

i.e. fO(x) ≤ fO(y). Below, we will only prove statements (1), (3) and (4), since(2) and (5) are order duals of (1) and (4).

(1). Let F ∈ F L; the set F ′ ∈ F L | eFL (F ′) ≤ eFL (F ) has a maximalelement, viz. F . It follows that

fO(eFL (F )

)=∨eFM F f(F ′) | eFL (F ′) ≤ eFL (F ) by definition of fO,

= eFM F f(F ) since eFM F f is order-preserving.

(3). Let x ∈ Lδ. Recall that fO(x) =∨eFM F f(F ) | eFL (F ) ≤ x and

fM(x) =∧eIM I f(I) | x ≤ eIL(I); we now have to show that∨eFM F f(F ) | eFL (F ) ≤ x ≤

∧eIM I f(I) | x ≤ eIL(I). (2.7)

Take F ∈ F L, I ∈ I L such that eFL (F ) ≤ x ≤ eI(I), then it follows bycompactness that F G I. Now by Lemma 2.1.2, we get that F f(F ) G I f(I), sothat also eFM F f(F ) ≤ eIM I f(I). It follows that (2.7) holds.

(4). Let I ∈ I L. We claim that

∀J ∈ IM,[∀F ∈ F L, F G I ⇒ F f(F ) G J

]iff I f(I) ⊆ J. (2.8)

Take J ∈ IM and suppose that the left-hand side of (2.8) holds. Take a ∈ I,then ↑ a G I, so by our assumption regarding J , ↑ f(a) G J , i.e. f(a) ∈ J . Itfollows that f [I] ⊆ J and thus I f(I) = ↓ f [I] ⊆ J . Conversely, suppose thatI f(I) ⊆ J and that F ∈ F L such that F G I. Then there is some a ∈ F ∩ I.Since a ∈ F , we also get f(a) ∈ f [F ] ⊆ ↑ f [F ] = F f(F ). Since a ∈ I, we get thatf(a) ∈ f [I] ⊆ ↓ f [I] = I f(I). Since we assumed that I f(I) ⊆ J , it follows thatf(a) ∈ J . But then F f(F ) G J . It follows that (2.8) holds.


We can now see that

fO(eIL(I)

)=∨eFM F f(F ) | eFL (F ) ≤ eIL(I) by def. of fO,

=∨eFM F f(F ) | F G I by compactness,

=∧eIM(J) | ∀F

[F G I ⇒ F f(F ) G J

] by Lemma 2.1.29,

=∧eIM(J) | I f(I) ⊆ J by (2.8),

= eIM I f(I) by order theory,

= fM(eIL(I)

)by (2).

It is known that the lower extension of an arbitrary map f : L→M betweendistributive lattices L, M can be characterized as the largest continuous extensionof f [39, Theorem 2.15]. The result below, which is new, tells us that we can saythe same about maps between non-distributive lattices, if we assume that f isorder-preserving. We will return to this issue in §3.2.2.

2.2.4. Theorem. Let f : L→M be an order-preserving map between lattices.

1. (a) The map fO : Lδ →Mδ is (δ↑, σ↑)-continuous and fO eL = eM f .

(b) The map fM : Lδ →Mδ is (δ↓, σ↓)-continuous and fM eL = eM f .

2. Let f ′ : Lδ →Mδ be an order-preserving extension of f : L→M, i.e. assumethat f ′ eL = eM f .

(a) If f ′ is (δ↑, ι↑)-continuous, then f ′ ≤ fO.

(b) If f ′ is (δ↓, ι↓)-continuous, then fM ≤ f ′.

Proof We will only prove the statements concerning fO, since the proofs for thoseconcerning fM are identical modulo order duality.

(1). Let x ∈ Lδ; we will first show that fO is locally (δ↑, σ↑)-continuous at x.Suppose that fO(x) ∈ U , where U ⊆Mδ is a σ↑-open set. Recall that

fO(x) =∨eFM F f(F ) | eFL (F ) ≤ x,

and observe that this join is directed by Lemma 2.1.28(1). Since fO(x) ∈ U ,it follows by definition of the σ↑-topology that some element of the join abovelies in U , i.e. that there must be some F ∈ F L such that eFL (F ) ≤ x andeFM F f(F ) ∈ U . Now ↑ eFL (F ) is a δ↑-open neighborhood of x; it remains to beshown that fO[↑ eFL (F )] ⊆ U . But this is easy to see: since fO is order-preserving,it follows by Fact A.3.1 that fO[↑ eFL (F )] ⊆ ↑ fO(eFL (F )). Now by Lemma 2.2.3(1),fO(eFL (F )) = eFM(F ) F f(F ); since we assumed that eFM(F ) F f(F ) ∈ U andwe know that U is an upper set, it follows that ↑ fO(eFL (F )) ⊆ U . Since U wasarbitrary, it follows that fO is locally (δ↑, σ↑)-continuous at x; since x ∈ Lδ wasarbitrary, it follows that fO is (δ↑, σ↑)-continuous.


To see that fO eL = eM f , observe that the right square below commutesby Lemma 2.2.3(2) and the left square commutes by the universal property of F .

L↑L //

f

F LeFL //

F f

Lδ

fO

M ↑M

// FMeFM

//Mδ

(2). Let f ′ : Lδ →Mδ be an order-preserving extension of f : L→M. We willshow something stronger than statement (2)(a): we will show that for all x ∈ Lδ,if f ′ is locally (δ↑, ι↑)-continuous at x, then f ′(x) ≤ fO(x). Suppose towards acontradiction that f ′ is locally continuous but that we have x ∈ Lδ such thatf ′(x) fO(x). Now since ↓ fO(x) is ι↑-closed, by local continuity there must besome δ↑-open set U ⊆ Lδ such that x ∈ U and f ′[U ] ⊆ Mδ \ ↓ fO(x). We mayassume that U is a basic open set, i.e. that U = ↑ eFL (F ) for some F ∈ F L, so itmust be the case that eFL (F ) ≤ x and f ′(eFL (F )) /∈ ↓ fO(x). But now we can usethe fact that f ′ is order-preserving to see that

f ′(eFL (F )

)= f ′(

∧eL[F ]) by definition of eFL ,

≤∧f ′ eL[F ] since f ′ is order-preserving,

=∧eM f [F ] because f ′ eL = eM f ,

= eFM F f(F ) by Remark 2.2.2,

≤ fO(x) by definition of fO.

This is a contradiction because we also assumed that f ′(eFL (F )) /∈ ↓ fO(x); itfollows that indeed, local continuity of f ′ at x implies that f ′(x) ≤ fO(x). Thus,if f ′ is continuous at every x ∈ Lδ, it follows that f ′ ≤ fO.

Even though fO is not Scott-continuous in general, it is when we restrict it toideal elements. The following result is known from the study of duality theory forHeyting algebras and modal algebras, but it holds in fact at the level of generalityof lattices and order-preserving maps.

2.2.5. Corollary (Esakia’s Lemma). Let f : L→M be an order-preservingmap between lattices L,M. Then fO eIL = fM eIL is Scott-continuous andfM eFL = fO eFL is co-Scott continuous.

Proof We only consider the first statement. The equality follows by Lemma 2.2.3;the continuity follows since eIL is (σ↑, δ↑)-continuous by Lemma 2.1.20 and fO is(δ↑, σ↑)-continuous by Theorem 2.2.4.

As another application of Lemma 2.2.3, we will show that the canonicalextension of an order embedding is again an order embedding.


2.2.6. Lemma. Let L, M be lattices. If f : L→M is an order embedding, thenso are fO and fM.

Proof Let f : L→M be an order embedding. We will only prove part (1); part(2) follows by order duality. Let x, y ∈ Lδ and suppose that fO(x) ≤ fO(y). Wewill show that

∀F ∈ F L, ∀I ∈ I L, if eFL (F ) ≤ x and y ≤ eIL(I), then eFL (F ) ≤ eIL(I). (2.9)

This is not hard to see. Take F ∈ F L and I ∈ I L such that eFL (F ) ≤ x andy ≤ eIL(I). Then

eFM F f(F ) = fO(eFL (F )

)by Lemma 2.2.3 (1),

≤ fO(x) since eFL (F ) ≤ x,

≤ fO(y) by assumption,

≤ fO(eIL(I)

)since y ≤ eIL(I),

= eIM I f(I) by Lemma 2.2.3 (2).

Consequently, F f(F ) G I f(I), i.e. ↑ f [F ] G ↓ f [I]. It follows that there must exista ∈ F and b ∈ I such that f(a) ≤ f(b). Since f is an order embedding, a ≤ b,so F G I. It follows that eFL (F ) ≤ eIL(I), concluding our proof of (2.9). It nowfollows from the density of filter and ideal elements that x ≤ y, concluding ourproof.

2.2.2 Operators and join-preserving maps

So far we have seen how to take an order-preserving map f : L→M and extend itcovariantly to maps F f : F L→ FM and I f : I L→ IM at the intermediatelevel of filters and ideals, and from there to maps fO : Lδ →Mδ and fM : Lδ →Mδ.In this section we will see that if f : L → M preserves binary joins, then theset-theoretic inverse function f−1 : P(L)→ P(M) becomes a partial function fromIM to I L. This contravariant partial extension of f at the intermediate levelwill then tell us a lot about the topological properties of fO : Lδ →Mδ.

Applying join-preserving maps to filters and ideals

We begin by making the observation that when an order-preserving map g : L→Mpreserves binary joins, we not only get a map I g : I L → IM, defined asI g(I) := ↓ g[I], but also a well-behaved partial map g−1 : IM→ I L.

2.2.7. Lemma. Let L, M be lattices and let g : L→M be a map preserving binaryjoins. Then

1. ∀J ∈ IM, g−1(J) ∈ I L iff g−1(J) 6= ∅;


2. ∀F ∈ F L, ∀J ∈ IM, F g(F ) G J iff g−1(J) ∈ I L and F G g−1(J).

Proof (1). Let J ∈ IM. Since ideals are non-empty by definition, the left-to-right implication is immediate. For the converse, first observe that g−1(J) isnon-empty by assumption. Moreover, g−1 is order-preserving, so by Fact A.3.1(3),g−1(J) is a lower set. Moreover, if a, b ∈ g−1(J), then g(a), g(b) ∈ J , so since Jis an ideal we get g(a) ∨ g(b) ∈ J . Since g preserves binary joins, we see thatg(a ∨ b) = g(a) ∨ g(b) ∈ J , so that a ∨ b ∈ g−1(J). It follows that g−1(J) is anideal of L.

(2). Let F ∈ F L and J ∈ IM. If F g(F ) G J , then ↑ g[F ] G J , i.e. theremust be some b ∈ (↑ g[F ]) ∩ J . Since b ∈ ↑ g[F ], there is some a ∈ F such thatg(a) ≤ b. Since b ∈ J and J is a lower set, it follows that g(a) ∈ J . But thena ∈ F ∩ g−1(J), so that F G g−1(J). Moreover, since g−1(J) 6= ∅, it follows by (1)that g−1(J) ∈ I L. Conversely, if F G g−1(J) then there must exist a ∈ F ∩g−1(J),i.e. a ∈ F and g(a) ∈ J . Since g(a) ∈ g[F ] ⊆ ↑ g[F ], it follows that ↑ g[F ] G J .

Up to this point, we have only considered canonical extensions of unary mapsf : L→M. We would now like to state several more detailed results, concerningcanonical extensions of n-ary maps. This is non-problematic since canonicalextensions commute with finite products; however, we would like to ignore thetechnical difference between, say, (L1 × L2)δ and Lδ1 × Lδ2 whenever possible.

2.2.8. Convention. If f : L1 × · · · × Ln → M is an n-ary order-preservingmap, then we regard F f as a map F L1 × · · · × F Ln → FM and fO as amap Lδ1 × · · · × Lδn → Mδ, rather than as maps F(L1 × · · · × Ln) → FM and(L1 × · · · × Ln)δ →Mδ. This is justified by Fact A.5.4 and Lemma 2.1.26. Usingthis convention, fO is calculated as follows for (x1, . . . , xn) ∈ Lδ1 × · · · × Lδn:

fO(x1, . . . , xn) =∨eFM F f(F1, . . . , Fn) | eFL1

(F1) ≤ x1, . . . , eFLn(Fn) ≤ xn,

and fM(x1, . . . , xn) is calculated similarly.

We will now work towards the main technical lemma of this section, whichconcerns canonical extensions of binary maps f : L1 × L2 → M which preservejoins in only one coordinate. We will first prove a result about the extensionF f : F L1 × F L2 → FM of such a map f , which will also be of use to us in§2.3.3.

2.2.9. Lemma. Let f : L1 × L2 →M be an order-preserving map between latticesL1, L2 and M which preserves binary joins in its first coordinate. Let S ∪ F ′ ⊆F L1 and G ∈ F L2. If S 6= ∅ and

∀I ∈ I L1, [∀F ∈ S, F G I]⇒ F ′ G I, (2.10)


then also

∀J ∈ IM, [∀F ∈ S, F f(F,G) G J ]⇒ F f(F ′, G) G J. (2.11)

Proof For each b ∈ L2, we define fb : L1 →M as fb : a 7→ f(a, b). It follows fromour assumptions above that for each b ∈ L2, fb : L1 → M is a map preservingbinary joins. Now we claim that

∀b ∈ G, ∀F ∈ F L1, F f(F,G) ⊇ F fb(F ). (2.12)

After all,

F f(F,G) = ↑ f [F,G] by definition of F f ,

⊇ ↑ f [F, b] since G ⊇ b,= ↑ fb[F ] by definition of fb,

= F fb(F ). by definition of F fb.

Now suppose that (2.10) holds and that J ∈ IM such that ∀F ∈ S, F f(F,G) G J .We need to show that F f(F ′, G) G J . We define I1 :=

⋃b∈Gf

−1b (J). We claim

thatI1 ∈ I L1, i.e. I1 is an ideal of L1. (2.13)

To establish this we need to show three things: we want that I1 is a lower set,that I1 is directed and that I1 is non-empty. Since fb is order-preserving for eachb ∈ G, it follows by Fact A.3.1(3) that I1 is a union of lower sets and hence, itselfa lower set. To see that I1 is directed, first observe that

for all b, b′ ∈ G, if b ≥ b′ then f−1b (J) ⊆ f−1

b′ (J). (2.14)

After all, if a ∈ f−1b (J), then f(a, b) ∈ J . Since f is order-preserving and b′ ≤ b,

we see that f(a, b′) ≤ f(a, b) ∈ J . Since J is a lower set, it follows that f(a, b′) ∈ J ,so that a ∈ f−1

b′ ; since a ∈ f−1b was arbitrary, it follows that (2.14) holds. Now

since G is a filter, it is co-directed; consequently, I1 :=⋃b∈Gf

−1b (J) is a directed

union. To see why I1 is a directed subset of L1, consider a, a′ ∈ I1. Because I1

is a directed union, there must exist some b ∈ G such that a, a′ ∈ f−1b (J). Now

f−1b (J) is non-empty, so it is an ideal by Lemma 2.2.7(1); consequently, there must

exist some c ∈ f−1b (J) such that a, a′ ≤ c. Since f−1

b (J) ⊆ I1, it follows that I1 isdirected. Finally, to see that I1 is non-empty, observe that since S is non-empty,there is some F ∈ S such that F f(F,G) G J . Since F f(F,G) := ↑ f [F,G], thismeans that there is some c ∈ (↑ f [F,G]) ∩ J . Since c ∈ ↑ f [F,G], there must bea ∈ F and b ∈ G such that f(a, b) ≤ c. Since J is a lower set and c ∈ J , it followsthat f(a, b) ∈ J . But then also fb(a) ∈ J , so that a ∈ f−1

b (J); since f−1b (J) ⊆ I1,

it follows that I1 6= ∅. We conclude that (2.13) holds. Next, we observe that

If ∀F ∈ S, F f(F,G) G J , then ∀F ∈ S,∃b ∈ G, F G f−1b (J). (2.15)


Suppose that the left-hand side of (2.15) holds and take F ∈ S. Then F f(F,G) GJ , so as we have seen above, there must exist a ∈ F and b ∈ G such thata ∈ f−1

b (J). It follows that a ∈ F ∩ f−1b (J), so that F G f−1

b (J). Since F ∈ S wasarbitrary, it follows that (2.15) holds. Recall that we assumed that J ∈ IM suchthat ∀F ∈ S, F f(F,G) G J ; we now see that

∀F ∈ S, F f(F,G) G J by assumption,

⇒ ∀F ∈ S,∃b ∈ G, F G f−1b (J) by (2.15),

⇒ ∀F ∈ S, F G I1 since I1 =⋃Gf−1b (J),

⇒ F ′ G I1 by (2.13) and (2.10),

⇒ ∃b ∈ G, F ′ G f−1b (J) by def. of I1,

⇒ ∃b ∈ G, F fb(F ′) G J by Lemma 2.2.7(2),

⇒ F f(F,G) G J by (2.12) and L. 2.1.2(1).

Since J ∈ IM was arbitrary, it follows that (2.11) holds.

The following well-known lemma is perhaps the most powerful technical resultin the theory of canonical extensions of maps between lattices.

2.2.10. Lemma ([34]). Let e1 : L1 → Lδ1, e2 : L2 → Lδ2 and eM : M → Mδ becanonical extensions of lattices, and let f : L1 × L2 → M be a order-preservingmap which preseves binary joins in the first coordinate. Then fO : Lδ1 × Lδ2 →Mδ

preseves all non-empty joins in the first coordinate.

Proof Let T ⊆ Lδ1 be a non-empty set and let y ∈ Lδ2. To show that fO(∨T, y) =∨

x∈TfO(x, y), it suffices to show that

fO(∨T, y) ≤

∨x∈Tf

O(x, y), (2.16)

since fO is order-preserving. Using the definition of fO, one can show that

fO(∨T, y) =

∨fO(

∨T, eF2 (G)) | eF2 (G) ≤ y,

and that ∨x∈Tf

O(x, y) =∨∨x∈Tf

O(x, eF2 (G)) | eF2 (G) ≤ y.

Thus we see that it suffices to show that for arbitrary G ∈ F L2,

fO(∨T, eF2 (G)) ≤

∨x∈Tf

O(x, eF2 (G)). (2.17)

Fix G ∈ F L2 and define S := F ∈ F L1 | ∃x ∈ T, eF1 (F ) ≤ x; we see that∨F∈Se

F1 (F ) =

∨T . Now if we look at the left-hand side of (2.17) then we see that

fO(∨T, eF2 (G))

= fO(∨F∈Se

F1 (F ), eF2 (G))

=∨eFM F f(F ′, G) | eF1 (F ′) ≤

∨F∈Se

F1 (F ) by def. of fO.


The right-hand side of (2.17) reduces as follows:∨x∈Tf

O(x, eF2 (G))

=∨x∈T∨eFM F f(F,G) | eF1 (F ) ≤ x by def. of fO,

=∨eFM F f(F,G) | F ∈ S by def. of S,

=∧eIM(J) | ∀F ∈ S, F f(F,G) G J by L. 2.1.29.

Thus we see that to show that (2.17) holds, it suffices to show that for allF ′ ∈ F L1 such that eF1 (F ′) ≤

∨F∈Se

F1 (F ) and for all J ∈ IM such that

∀F ∈ S, F f(F,G) G J , we have that

eFM F f(F ′, G) ≤ eIM(J). (2.18)

Now given the fact that eF1 (F ′) ≤∨F∈Se

F1 (F ), we know from Lemma 2.1.29 that

eF1 (F ′) ≤∧eI1 (I) | ∀F ∈ S, F G I,

i.e. for all I ∈ I L1, if ∀F ∈ S, F G I, then F ′ G I. But now it follows from Lemma2.2.9 and our assumption about J that F f(F ′, G) G J , so that (2.18) holds. SinceF ′ and J were arbitrary, it follows that (2.17) holds, which concludes our proof.

Operators and dual operators

Obviously, we can use Lemma 2.2.10 to make claims about join-preserving maps,but that is not all. Operators form another example of maps which satisfy theconditions of Lemma 2.2.10.

2.2.11. Definition. Let f : L1 × · · · × Ln → M be an n-ary order-preservingmap between lattices. We call f an operator if f preserves binary joins in eachcoordinate, i.e. if for all i ≤ n, for all a1, . . . , an ∈ L1 × · · · × Ln and all b ∈ Li,we have

f(a1, . . . , ai ∨ b, . . . , an) = f(a1, . . . , ai, . . . , an) ∨ f(a1, . . . , b, . . . , an).

If all lattices involved are complete and if f preserves all non-empty joins in eachcoordinate, then we call f a complete operator.

We call f a normal operator if f is an operator and for all i ≤ n, for alla1, . . . , an ∈ L1 × · · · × Ln,

ai = 0⇒ f(a1, . . . , ai, . . . , an) = 0.

In other words, f is a normal operator if it also preserves the empty join in eachcoordinate.

A dual operator (complete dual operator, etc.) is an n-ary map which preservesbinary meets (all non-empty meets, etc.) in each coordinate.


Operators arise, for instance, in the algebraic semantics for modal logics [19,Ch. 5], where they correspond to existential modalities (usually denoted by ‘♦’).

2.2.12. Example. The property of being a (normal) operator is weaker thanthat of being a join-homomorphism. Consider the map f : 0, 1×0, 1 → 0, 1,with the usual order on 0, 1, defined as

f : (a, b) 7→ a ∧ b.

Then f is an operator, in fact a normal operator, because 0, 1 is a distributivelattice. However, f is not a join-homomorphism:

f((0, 1) ∨ (1, 0)) = f(1, 1) = 1 ∧ 1 = 1,

but

f(0, 1) ∨ f(1, 0) = (0 ∧ 1) ∨ (1 ∧ 0) = 0 ∨ 0 = 0.

This is quite different from the situation for directed joins; if g : D1×· · ·×Dn →E is a map between dcpo’s which preserves directed joins in each coordinate, thenby Fact A.3.4, g preserves directed joins in D1 × · · · × Dn.

In Lemma 2.2.10, we only considered binary joins, i.e. non-empty finite joins.Canonical extensions also behave well with respect to maps which preserve theempty join.

2.2.13. Lemma. Let e1 : L1 → Lδ1, e2 : L2 → Lδ2 and eM : M → Mδ be canonicalextensions of lattices, and let f : L1 × L2 →M be a order-preserving map.

1. If ∀b ∈ L2, f(0, b) = 0, then also ∀y ∈ Lδ2, fO(0, y) = 0;

2. If ∀b ∈ L2, f(1, b) = 1, then also ∀y ∈ Lδ2, fO(1, y) = 1;

Proof We will only prove (1), since (2) is just the order dual of (1). First, observethat

∀F ∈ F L1, ∀G ∈ F L2, 0 ∈ F ⇒ 0 ∈ F f(F,G). (2.19)

Take F ∈ F L1 and G ∈ F L2 such that 0 ∈ F , then since G must be non-empty,there is some b ∈ G. Now

f(0, b) ∈ f [F,G] ⊆ ↑ f [F,G] = F f(F,G),

so since f(0, b) = 0 by assumption, we see that 0 ∈ F f(F,G). Next, observe thatit is a basic fact about canonical extensions that for any lattice L,

∀a ∈ L,∀F ∈ F L, a ∈ F iff eFL (F ) ≤ eL(a). (2.20)


After all, a ∈ F iff F G ↓ a iff eFL (F ) ≤ eIL(↓ a) = eL(a). Now we can see that forany y ∈ Lδ2,

fO(0, y)

= fO(e(0), y) since e(0) = 0,

=∨eM F f(F,G) | eF1 (F ) ≤ e1(0), eF2 (G) ≤ y by def. of fO,

=∨eM F f(F,G) | 0 ∈ F, eF2 (G) ≤ y by (2.20),

≤∨eM(F ′) | 0 ∈ F ′ by (2.19),

=∨eM(F ′) | eM(F ′) ≤ e(0) by (2.20),

= 0,

which is what we wanted to show.

We can now state an immediate corollary of Lemmas 2.2.10 and 2.2.13:

2.2.14. Corollary ([34]). Let f : L1 × · · · × Ln → M be an n-ary order-preserving map between lattices.

1. If f is a (normal) operator, then fO is a complete (normal) operator.

2. If f is a dual (normal) operator, then fM is a complete dual (normal)operator.

Topological properties of operators and join-preserving maps

It is one of the characteristic features of canonical extension that maps betweenlattices (f : L→M ) have both a lower (fO : Lδ →Mδ) and an upper extension(fM : Lδ →Mδ). These two extensions need not necessarily be different.

2.2.15. Definition. We say that an order-preserving map f : L→M is smoothif fO = fM. If we want to emphasize that f is smooth, we will refer to the canonicalextension of f as f δ rather than fO or fM.

In light of the recurring topological themes in this chapter, it may not comeas a surprise that smoothness of a map f : L→M is a topological property.

2.2.16. Lemma. an order-preserving map f : L→M between lattices is smoothiff fO : Lδ →Mδ is (δ, σ)-continuous.

Proof If fO = fM, then fO is both (δ↑, σ↑)-continuous and (δ↓, σ↓)-continuous;hence fO is also (δ, σ)-continuous. Conversely, if fO is (δ, σ)-continuous, then itfollows from Lemma 2.1.17 and the fact that fO is order-preserving that fO is(δ↓, σ↓)-continuous. It follows by the order dual of Theorem 2.2.4 that fM ≤ fO.Since fO ≤ fM by Lemma 2.2.3, we find that fO = fM.


Before we proceed to the main result about topological properties of operatorsand join-preserving maps, we prove another technical lemma which says, intuitu-ively, that if f : L→M preserves binary joins, then fO : Lδ →Mδ has a kind ofweak, partial right adjoint.

2.2.17. Lemma. Let f : L → M be an order-preserving map preserving binaryjoins.

∀x ∈ Lδ,∀J ∈ IM, fO(x) ≤ eIM(J) iff f−1(J) ∈ I L and x ≤ eIL f−1(J).

Proof Let x ∈ Lδ and J ∈ IM. We define S := F ∈ F L | eFL (F ) ≤ x; observethat S is always non-empty since at least ↑ 0 ∈ S. Now if fO(x) ≤ eIM(J), thensince fO(x) =

∨F∈Se

FM F f(F ), we see that ∀F ∈ S, eFM F f(F ) ≤ eIM(J). By

basic properties of canonical extension it follows that ∀F ∈ S,F f(F ) G J . SinceS 6= ∅, there is at least one F ∈ F L such that F f(F ) G J , so by Lemma 2.2.7(2),f−1(J) ∈ I L. It also follows by Lemma 2.2.7(2) that ∀F ∈ S, F G f−1(J),so that ∀F ∈ S, eFL (F ) ≤ eIL f−1(J). Since x =

∨F∈Se

FL (F ), it follows that

x ≤ eIL f−1(J). The proof of the converse implication is analogous.

We now arrive at the main theorem about topological properties of operatorsand join-preserving maps. Parts (1) and (3) were already known from [34].

2.2.18. Theorem. Let f : L→M be an order-preserving map between lattices.

1. (a) If f is an operator, then fO is (σ↑, σ↑)-continuous.

(b) If f is a dual operator, then fO is (σ↓, σ↓)-continuous.

2. (a) If f preserves binary joins, then fO : Lδ →Mδ is (δ↓, δ↓)-continuous.

(b) If f preserves binary meets, then fM : Lδ →Mδ is (δ↑, δ↑)-continuous.

3. If f preserves binary joins or binary meets, then f is smooth.

4. If f preserves binary joins and binary meets, then f is smooth and f δ : Lδ →Mδ is (δ, δ)-continuous.

Proof We will only show the proofs for the statements about fO, since the proofsfor the statements about fM are order dual.

(1). By Lemma 2.2.10, fO preserves all non-empty joins in each coordinate,so a fortiori fO preserves directed joins in each coordinate. It follows from FactA.3.4 that fO : Lδ1 × · · · × Lδn →Mδ preserves directed joins.

(2). We will show that (fO)−1 maps basic δ↓-open sets to δ↓-open sets. LetJ ∈ IM; we need to show that (fO)−1

(↓ eIM(J)

)is δ↓-open. If (fO)−1

(↓ eIM(J)

)is empty then we are done. If not, then it follows from Lemma 2.2.17 that

(fO)−1(↓ eIM(J)

)= ↓ eIL

(f−1(J)

), (2.21)


so we see that (fO)−1(↓ eIM(J)

)is in fact a basic δ↓-open set.

(3). Suppose that f preserves binary joins; then by (2), fO : Lδ → Mδ

is (δ↓, δ↓)-continuous. Since σ↓ ⊆ δ↓ (Lemma 2.1.28(3), it follows that fO is(δ↓, σ↓)-continuous. On the other hand, by Theorem 2.2.4(1) we know that fO is(δ↑, σ↑)-continuous. It now follows from Lemma 2.2.16 that f is smooth.

(4). It follows from (3) that f is smooth. Now by (2), fO is (δ↓, δ↓)-continuousand fM is (δ↑, δ↑)-continuous. Since f is smooth, i.e. since fO = fM, it now followsfrom general topology that f δ := fO is (δ, δ)-continuous.

We now turn to an interesting question which we have neglected so far. Wetook it as part of our definition that eL : L → Lδ is a lattice embedding, whichmeans that the meet and join of Lδ a priori ‘play nice’ with those of L. If welook at meet and join as maps ∨L : L× L→ L and ∧L : L× L→ L however, wecan also ask ourselves what are the canonical extensions of ∨L and ∧L. Are theseindeed the join and meet of Lδ? Fortunately, the answer is yes.

2.2.19. Lemma ([34]). Let e : L→ Lδ be a canonical extension. Then (∨L)O =(∨L)M = ∨Lδ and (∧L)O = (∧L)M = ∧Lδ .

Proof We will only consider ∨, since the other case follows by order duality. Since∨L : L× L→ L is associative, it is a join-preserving map, so by Theorem 2.2.18(∨L)O = (∨L)M. It follows from order theory that for all x, y ∈ Lδ,

x ∨Lδ y =∧z ∈ Lδ | x, y ≤ z.

By meet-density of ideal elements, this reduces to

x ∨Lδ y =∧I∈Se

I(I),

where S := I ∈ I L | x, y ≤ eI(I). On the other hand,

x(∨L)My =∧J∈S′e

I(J),

where S ′ := I ∨L(J1, J2) | x ≤ eI(J1), y ≤ eI(J2). Since I ∨L = ∨I L, we seethat if I ∈ S then I ∨L(I, I) = I ∈ S ′, so S ′ ⊆ S. Conversely, if I ∨L(J1, J2) ∈ S ′and x ≤ eI(J1), y ≤ eI(J2), then also x, y ≤ eI(I ∨L(J1, J2)), so I ∨L(J1, J2) ∈ Sand hence S ′ ⊆ S. It follows that (∨L)M = ∨Lδ .

Now if L is a distributive lattice, then we know that ∧L : L × L → L is anoperator. Consequently, by Corollary 2.2.14, (∧L)O = ∧δL is a complete operator,so as a bonus we get the following well-known corollary:

2.2.20. Corollary. If L is a distributive lattice then so is Lδ.

2.2.21. Remark. Canonical extensions of distributive lattices have much strongerproperties than just being distributive. We will return to this subject in §4.1.


2.2.3 Canonical extension as a functor I: lattices only

So far in this chapter we have seen that canonical extension is a construction onlattices and maps between lattices. This raises the very natural question whethercanonical extension is a functor, and if so between which categories. In thissubsection we will see that canonical extension is a functor from the category ofbounded lattices and lattice homomorphisms to the category of complete latticesand complete homomorphisms.

In order to establish this result, we will first prove several facts about theinteraction between canonical extensions and compositions of order-preservingmaps. The most basic such result is the well-known fact [34] that if we have twoorder-preserving maps

L1f // L2

g // L3 ,

then it is always the case that (gf)O ≤ gOfO and dually, gMfM ≤ (gf)M. Thisresult is supplemented by the observation that if we make certain continuityassumptions about fO or gO, we can prove the reverse inequality. This fact wasalready known in the case of order-preserving maps between distributive lattices[39], however the result is new for the non-distributive case. Armed with theseresults we can then prove Theorem 2.2.24, which says that canonical extensionis a functor from the category of lattices to the category of complete lattices.Theorem 2.2.24 extends a known result from [34] with a new observation aboutthe continuity properties of canonical extensions of lattice homomorphisms. Wewill revisit the subject of compositions of canonical extensions of maps in §3.2.3,and the subject of functorial behaviour of canonical extension in §3.3.

We will now first state several results about canonical extensions of composi-tions of order-preserving maps. We begin with a well-known result.

2.2.22. Lemma ([34]). Let ei : Li → Lδi be canonical extensions of lattices L1,L2,L3

and let f : L1 → L2 and g : L2 → L3 be order-preserving maps. Then the followinginequalities hold:

(gf)O ≤ gOfO ≤gOfM

gMfO

≤ gMfM ≤ (gf)M.

Proof The inequalities gOfO ≤ gOfM and gOfO ≤ gMfO follow from Lemma 2.2.3.For the first inequality, observe that

(gf)O(x) =∨eF3 F gf(F ) | eF1 (F ) ≤ x =∨

eF3 F g F f(F ) | eF1 (F ) ≤ x ≤∨eF3 F g(F ′) | eF2 (F ′) ≤ fO(x),

where the inequality directly above follows from the fact that if eF1 (F ) ≤ x, thenalso

eF2 F f(F ) = fO(eF1 (F )

)≤ fO(x).

The other inequalities in the statement of the lemma follow by order duality.


Next, we present a handful of corollaries of the above lemma.

2.2.23. Corollary. Let f : L1 → L2 and g : L2 → L3 be order-preserving mapsbetween lattices.

1. If gOfO is (δ↑, σ↑)-continuous then gOfO = (gf)O;

2. If gO is (σ↑, σ↑)-continuous then gOfO = (gf)O;

3. If fO is (δ↑, δ↑)-continuous then gOfO = (gf)O.

Proof (1). If gOfO is (δ↑, σ↑)-continuous then by Theorem 2.2.4, gOfO ≤ (gf)O.By Lemma 2.2.22, (gf)O ≤ gOfO. Statements (2) and (3) are instances of (1).

Recall from Definition 2.2.15 that we call a map f : L→M smooth if fO = fM,and that we refer to the canonical extension of f as f δ : Lδ → Mδ in that case.Additionally, recall from §A.4 that we denote the category of bounded latticesand lattice homomorphisms by Lat, and the category of complete lattices andcomplete lattice homomorphisms by CLat. We can now state a fundamentaltheorem about canonical extensions of lattices. Most of this theorem was alreadyknown, see e.g. [34]; part (1) is a new observation however.

2.2.24. Theorem. Let L, M be lattices and let f : L→M be a lattice homomor-phism. Then f is smooth and

1. f δ : Lδ → Mδ is a complete lattice homomorphism which is both (δ, δ)-continuous and (σ, σ)-continuous;

2. If f is injective, then so is f δ;

3. If f is surjective, then so is f δ.

In fact, canonical extension defines a functor from Lat to CLat and eL : L→ Lδis a natural transformation.

Proof (1). Since f preserves binary joins and binary meets, it follows by Lemma2.2.10 that f δ preserves all non-empty joins and meets. It follows from Lemma2.2.13 that f δ preserves 0 and 1. Now since complete homomorphisms preservedirected joins and co-directed meets a fortiori, it follows that f δ is (σ, σ)-continuous.Finally it follows from Theorem 2.2.18(2) that f δ is (δ, δ)-continuous.

(2). This follows from Lemma 2.2.6, since f preserves binary joins.


(3). Assume f is surjective; then by (1), f δ is a complete homomorphism, andwe already know that F f is surjective (see §A.5.1). Let x ∈Mδ; we now make astraightforward computation:

f δ(∨eFL (F ) | eFM F f(F ) ≤ x

)=∨f δ eFL (F ) | eFM F f(F ) ≤ x because f is a complete hom.,

=∨eFM F f(F ) | eFM F f(F ) ≤ x by Lemma 2.2.3,

=∨eFM(F ′) | eFM(F ′) ≤ x because F f is surjective,

= x by join-densite of filter elements.

Since x ∈Mδ was arbitrary it follows that f δ is surjective.To see that canonical extension is a functor, we will first look at compositions

of homomorphisms: consider lattice homomorphisms f : L1 → L2 and g : L2 → L3.Since gδ is Scott-continuous by (1), it follows by Corollary 2.2.23 that gδf δ = (gf)δ.To see that canonical extension preserves the identity function, we use that factthat F is a functor. Let x ∈ Lδ be arbitrary, then

(idL)δ(x) =∨eFL F idL(F ) | eFL (F ) ≤ x by definition,

=∨idF L eFL (F ) | eFL (F ) ≤ x because F is a functor,

=∨eFL (F ) | eFL (F ) ≤ x by definition of idF L,

= x by join-density of filter elements,

= idLδ(x)

This proves that canonical extension is a functor on lattices and lattice homomor-phisms. The fact that eL : L→ Lδ is a natural transformation follows from thefact that f δ is an extension of f .

We conclude this section with a minor observation about canonical extensionsof sublattices which will be useful later.

2.2.25. Corollary. If L is a sublattice ofM, then Lδ is isomorphic to a completesublattice of Mδ.


The main contribution of this section lies in the technical results concerning filtersand ideals, and the results about topological properties of fO and fM. Most of thetopological results in this section can also be found in our paper with M. Gehrke[43]. Many of these where inspired by what was known about distributive latticesfrom the work of Gehrke and Jonsson [39]. It was not known however thatdistributivity of the lattices involved, which is a central assumption in [39], is in

2.3. Canonical extensions via dcpo presentations 45

no way essential when one wants to discuss topological properties of canonicalextensions of order-preserving maps. In the work of Ghilardi and Meloni [44],the action of join-preserving maps on filters and ideals receives the attention itdeserves, although the authors nowhere refer explicitly to canonical extensions.The idea to define canonical extensions of order-preserving maps, rather thanjoin-preserving maps or operators, via the filter and ideal completion seems tohave been introduced by Gehrke and Priestley in [41], but in that paper mostattention was directed at lattice homomorphisms.

The definition of fO and fM for an order-preserving map in §2.2.1 is well knownfrom the work of Gehrke and Harding [34]. Theorem 2.2.4, which describes fO

and fM as the largest and smallest continuous extensions of an order-preservingmap f , respectively, was previously only known to hold for distributive lattices. Itraises questions which we will return to in Remark 3.2.22. The result concerningpreservation of order embeddings (Lemma 2.2.6) is also an improvement over whatwas previously known (viz. that the canonical extension of an injective latticehomomorphism is again injective).

The technical results about order-preserving maps applied to filters and idealsin §2.2.2 are new, although similar results can be found in [44]. It was alreadyknown from [34] that canonical extensions of join-preserving maps and operatorsare very well-behaved (Theorem 2.2.18); the topological results we presented arenew however.

Further work

• In this section, we have made much use of the filter completion and the idealcompletion, which we borrowed from domain theory. It would be interestingto see if there are more domain-theoretic tools available that we can use tobetter understand and develop the theory of canonical extensions. We willsee an example of this in the next section, where we use dcpo presentationsto describe canonical extensions.

• Another interesting question is to see whether the domain theory tools andtopological methods of this section can also be applied in the setting ofmonotone partially ordered algebras, as studied by Dunn et al. in [32].

• A more specific question which needs to be resolved is whether the canonicalextension of any order embedding is again an order embedding, cf. Lemma2.2.6.

2.3 Canonical extensions via dcpo presentations

Thus far, we have explored canonical extensions of lattices and order-preservingmaps via topological methods. We will conclude this chapter with a section, based


on results from [42], in which we show that canonical extensions can also beunderstood via dcpo presentations, a technique from domain theory [60]. Apartfrom being interesting in its own right, the perspective on canonical extensionsusing dcpo presentations sheds a different light on the issue of extending mapsbetween lattices to maps between their canonical extensions, and ultimately onthe question whether inequations valid on a distributive lattice with operators Aare also valid on its canonical extension Aδ, i.e. the question when inequations arecanonical.

The idea of dcpo presentations is to give a unique description of a dcpo D,that is, of a directed complete partial order, by specifiying a partially ordered orpre-ordered set of generators P such that each element of D is the directed join ofthe generators below it. If one imposes no relations on the generators other thanthe order, that is, if one freely adds all directed joins to P , then this is equivalentto taking the ideal completion of P . Thus we can give presentations of algebraicdcpos. If, on the other hand, one imposes relations of the shape a ≤

∨U , for

a ∪ U ⊆ P , then we may get a presentation for any dcpo, see [60].To see how we may use this to get a dcpo presentation of the canonical

extension of a lattice L, recall from Theorem 2.1.25 that there exists an orderembedding g : Lδ → I F L such that g eFL = ↓F L.

Lδg

##FFFFFFFFF

L ↑L//

eL

66mmmmmmmmmmmmmmmmm F L ↓F L

//eFL

==zzzzzzzzI F L

where g =(eFL)−1 ↓Lδ and eFL are embeddings. What this diagram tells us is

that the canonical extension of L ‘sits between’ F L and I F L, the structureone obtains by freely adding all directed joins to F L. The idea of the dcpopresentation of canonical extension is to be selective and only add some directedjoins to F L, so that we obtain Lδ. Because of the way that filters, ideals andcanonical extensions dualize with respect to order (see Lemma 2.1.26), we canjust as well regard Lδ as an object sitting in between I L and F I L; this wouldlead to a description of Lδ via a co-dcpo presentation over I L. We choose not toengage in this exercise of dualization.

Since dcpo presentations are characterized externally, that is, by conditions ontheir behaviour with respect to maps, it is quite natural to expect that we can usethem to describe canonical extensions of maps between lattices. In this section,we will restrict our attention to maps of the type f : Ln → L, with n a naturalnumber. This choice is dictated by economy rather than necessity: we will developjust enough of the technique of extending maps between lattices to canonicalextensions via dcpo presentations to allow us to prove a canonicity result in §3.3.3.The key observation is that under the right assumptions, the canonical extensionof a map can be seen as an instance of an extension via dcpo presentations, sothat one can apply results about dcpo presentations to canonical extensions.


2.3.1 Dcpo presentations

In this subsection, we introduce dcpo presentations, which are a technical toolfor uniquely specifying a dcpo without spelling out its entire structure. This isan instance of a very general algebra technique, namely that of specifications bygenerators and relations.

2.3.1. Definition. A dcpo presentation [60] is a triple 〈P,v, /〉 where

• 〈P,v〉 is a pre-order;

• / ⊆ P ×P(P ) is a binary relation such that a/U only if U ⊆ P is non-emptyand directed.

An order-preserving map f : P → D to a dcpo D is cover-stable if for all a / U ,f(a) ≤

∨f [U ].

In other words, a dcpo presentation consists of a pre-ordered set of generators〈P,v〉 together with set of relations of the form a ≤

∨U . But what does it mean

for a dcpo presentation to uniquely describe, i.e. to present a dcpo?

2.3.2. Definition. A dcpo presentation 〈P,v, /〉 presents a dcpo D if thereexists a cover-stable order-preserving map η : P → D such that for all dcpos E,if f : P → E is a cover-stable order-preserving map then there exists a uniqueScott-continuous f ′ : D→ E such that f ′ η = f . If this is the case, we say that〈P,v, /〉 presents D via η.

Df ′

P

η??~~~~~~~

f// E

We may ask ourselves if every dcpo presentation uniquely describes, i.e. presentsa dcpo. This is indeed the case; one can show this by constructing a dcpo froma given presentation using so-called C-ideals. For more information we refer thereader to [60].

2.3.3. Fact. Every dcpo presentation presents a dcpo.

We conclude this subsection with two trivial examples of dcpo presentations.

2.3.4. Example. If P = 〈P,≤〉 is a poset, then 〈P,≤, ∅〉 presents I P via ↓ : P→I P. This follows from the universal property of the ideal completion. Whatmakes this example trivial is the fact that there are no relations imposed on thegenerators.

If D is a dcpo, then 〈D,≤, /D〉, where a /D U iff a ≤∨U , presents D itself

via idD : D→ D. This is a trivial example because there are as many generators


as there are elements in the dcpo being presented and the order is already fullyspecified: the main reason for considering presentations of objects via generatorsand relations, rather than the objects themselves, is that the presentations can besimpler to describe than the objects they are presenting. In the current example,this is not the case.

2.3.2 A dcpo presentation of the canonical extension

We now define a dcpo presentation given a lattice L, with the aim of showing thatthis dcpo presentation presents Lδ. This is a two-stage process: first we take Land we define a presentation ∆(L), using F L as the set of generators. Then, weshow that ∆(L) presents Lδ.

2.3.5. Definition. Given a lattice L, we define a dcpo presentation ∆(L) :=〈F L,⊇, /L〉, where for all F ∈ F L and S ⊆ F L directed,

F /L S iff ∀I ∈ I L,[∀F ′ ∈ S, F ′ G I

]⇒ F G I.

We want to emphasize that this definition is not being pulled out of a hat. Firstly,since the filter elements of Lδ are join-dense in Lδ, it is not a strange idea totake the filters of L as generators. Secondly, we know by Lemma 2.1.29 that ife : L→ Lδ is the canonical extension of L, then∨

eF(F ) | F ∈ S =∧eI(I) | ∀F ∈ S, F G I,

and we will see in the proof of the theorem below that this equation is essentiallyequivalent to the relations of the shape F /L S we are imposing on ∆(L).

2.3.6. Theorem. Let L be a lattice and let e : L→ Lδ be its canonical extension.Then ∆(L) presents Lδ via eF : F L→ Lδ.

Proof Observe that eF : F L→ Lδ is order-preserving by Lemma 2.1.28(1); wenow first need to show that eF is cover-stable. We will show something stronger:for all F ∈ F L and S ⊆ F L directed,

F /L S iff eF(F ) ≤∨F ′∈S

eF(F ′). (2.22)

The key to this observation is Lemma 2.1.29, which states that∨F ′∈Se

F(F ′) =∧eI(I) | ∀F ′ ∈ S, F ′ G I. Now

eF(F ) ≤∨F ′∈Se

F(F ′)

iff eF(F ) ≤∧eI(I) | ∀F ′ ∈ S, F ′ G I by Lemma 2.1.29,

iff ∀I ∈ I L, [∀F ′ ∈ S, F ′ G I]⇒ eF(F ) ≤ eI(I) by order theory,

iff ∀I ∈ I L, [∀F ′ ∈ S, F ′ G I]⇒ F G I (†),iff F /L S by def. of /L,


where (†) follows from the basic fact about canonical extensions that eF(F ) ≤ eI(I)iff F G I. It follows that (2.22) holds.

Next, suppose that f : F L→ D is an order-preserving cover-stable map to adcpo D. We define f ′ : Lδ → D by setting

f ′ : x 7→∨f(F ) | eF(F ) ≤ x.

We need to show (1) that f ′ is well-defined and Scott-continuous, (2) that f ′eF = fand (3) that f ′ is unique with respect to properties (1) and (2).

(1). For any x ∈ Lδ, the set ↓x is an ideal. Since eF is a ∨-homomorphism(Lemma 2.1.28(1)), it follows that (eF)−1(↓x) = F | eF(F ) ≤ x is also an idealand hence, directed, Since f is order-preserving it follows that f ′(x) is a directedjoin, so that indeed f ′ is well-defined. To see that f ′ is Scott-continous, takeS ⊆ Lδ directed. Observe that∨

S

=∨x∈S

∨eF(F ) | eF(F ) ≤ x by join-density of filter elements,

=∨⋃

x∈SeF(F ) | eF(F ) ≤ x by associativity of

∨,

=∨

eF[⋃

x∈SF | eF(F ) ≤ x

]by elementary set theory.

It is not hard to see that⋃x∈SF | eF(F ) ≤ x is a directed union of directed

sets; consequently, we will simply assume that S is a directed set of filter elements;say S = eF(F ) | F ∈ S ′ where S ′ ⊆ F L is directed. Now observe that

f ′(∨

F∈S′eF(F )

)=∨

f(F ′) | eF(F ′) ≤∨F∈S′e

F(F )

=∨f(F ′) | F ′ /L S

′,

where the last equality follows by (2.22). Since f is cover-stable, F ′ /L S′ implies

f(F ′) ≤∨f [S ′]. We see that

f ′(∨

F∈S′eF(F )

)=∨f(F ′) | F ′ /L S

′ ≤∨F∈S′

f(F ) =∨F∈S′

f ′(eF(F )

),

so that it follows that f ′ is Scott-continuous.(2). To see that f ′ eF = f , observe that

f ′ eF(F ) =∨f(F ′) | eF(F ′) ≤ eF(F ).

Because eF is an order embedding, the join in the RHS above has a maximalelement, viz. f(F ). It follows that f ′ eF = f .

(3). Suppose that f ′′ : Lδ → D is Scott-continuous and that f ′′ eF = f . Takex ∈ Lδ, then since x =

∨eF(F ) | eF(F ) ≤ x is a directed join, we see that

f ′′(x) = f ′′(∨eF(F ) | eF(F ) ≤ x

)=∨f ′′ eF(F ) | eF(F ) ≤ x

=∨f(F ) | eF(F ) ≤ x = f ′(x).

It follows that f ′ = f ′′ so that f ′ is unique.


What have we learned now? Firstly, we have discovered a new characterizationof the canonical extension of a lattice L, namely as a certain dcpo generated bythe filters of L. Secondly, we have paved the way for applying general resultsabout dcpo algebras to canonical extensions of lattice-based algebras, as we willsee later.

2.3.3 Extending maps via dcpo presentations

We will now briefly look at extensions of maps via dcpo presentations. The goal isto be able to use results about dcpo presentations to prove facts about canonicalextensions. Specifically, we would like to be able to lift a map f : Ln → L throughour two-stage construction, first to a map defined on F L and then via the dcpopresentation to a map on Lδ. We first state the facts we need for the second stageof extending f .

Let 〈P,v, /〉 be a dcpo presentation and let f : P n → P be an order-preservingmap. We say f is cover-stable if f preserves covers in each coordinate, i.e. for all1 ≤ i ≤ n, for all a1, . . . , an ∈ P , for all U ⊆ P ,

ai / U ⇒ f(a1, . . . , an) / f(a1 . . . , ai−1, b, ai+1, . . . , an) | b ∈ U.

2.3.7. Fact ([60]). Let 〈P,v, /〉 be a dcpo presentation which presents a dcpo Dvia η : P → D. If f : P n → P is a cover-stable order-preserving map, then thereexists a unique Scott-continuous map f : Dn → D which extends f , i.e. such thatf ηn = η f .

Suppose we are given a map f : Ln → L, and recall that F f(F1, . . . , Fn) :=↑ f [F1, . . . , Fn]. How do we know if F f : (F L)n → F L is cover-stable? The bestcondition we know of is the rather strong requirement that f is an operator, i.e. iff preserves binary joins in each coordinate.

2.3.8. Lemma. Let L be a lattice. If f : Ln → L is an operator, then F f : (F L)n →F L is cover-stable with respect to ∆(L). Consequently, F f extends to a mapF f : (Lδ)n → Lδ; moreover, F f = fO.

Proof Let e : L→ Lδ be the canonical extension of L and let f : Ln → L be anoperator; to lighten the notation, we assume that n = 2.

First, we need to show that F f is cover-stable with respect to ∆(L). Itsuffices to show that F f is cover-stable in its first coordinate. So suppose thatF,G ∪ S ⊆ F L such that F /L S; we want to show that

F f(F,G) /L F f(F ′, G) | F ′ ∈ S.

By definition of /L, this amounts to showing that if

∀I ∈ I L1, [∀F ′ ∈ S, F G I]⇒ F G I,


then also

∀J ∈ IM, [∀F ′ ∈ S, F f(F ′, G) G J ]⇒ F f(F,G) G J.

But that is exactly the statement of Lemma 2.2.9, so it follows immediately thatF f is cover-stable.

Now, observe that by Fact 2.3.7, we know that F f has a Scott-continuousextension F f such that F f (eF × eF) = eF F f . We will show that F f = fO.Take x, y ∈ Lδ, then

F f(x, y) = F f(∨eF(F ) | eF(F ) ≤ x,

∨eF(G) | eF(G) ≤ y

),

by join-density of filter elements. Now since each of the joins above is directed(Lemma 2.1.28) and F f is Scott-continuous, we see that

F f(x, y) =∨F f(eF(F ), eF(G)) | eF(F ) ≤ x, eF(G) ≤ y

=∨

eF F f(F,G) | eF(F ) ≤ x, eF(G) ≤ y

by F. 2.3.7,

= fO(x, y),

where the last equality follows by definition of fO. Since x, y ∈ Lδ were arbitrary,it follows that F f = fO.

Thus, we see that canonical extensions of operators can be described usingdcpo presentation techniques. This concludes our discussion of dcpo presentationsfor now. We will use what we have learned here later on, in §3.3.3.


This section is based on a paper of M. Gehrke and the author [42], which waswritten to demonstrate how a canonicity result for distributive lattices withoperators from [38] can be seen as a special case of a result concerning dcpoalgebras from [60], see §3.3.3.

What we have seen here (and what we will see in §3.3.3) is part of theintersection of the structures and results which can be described both using dcpoalgebras and canonical extensions. It would be interesting to further explore theoverlapping area of the two fields.

Chapter 3

Canonical extensions: topologicalalgebra and categorical properties

In Chapter 2, we have presented a toolkit of results which allow us to workwith canonical extensions of bounded lattices and order-preserving maps, usingtechniques from domain theory. In this chapter, we will start looking at canonicalextensions of lattice-based algebras, which is the main application area of canonicalextensions, and we will see that canonical extensions are finely intertwined withtopological algebra.

As we discussed in Chapter 1, the main application of canonical extensions is toprovide representation theorems for lattice-based algebras, which arise in algebraiclogic. Via such representation theorems (e.g. the Jonsson-Tarski theorem [58]),questions about logics can be reduced to questions about canonical extensions,and this is where the canonical extensions toolkit can be very useful. (We dopoint out however that in this chapter we are concerned with the mathematicalproperties of canonical extensions, rather than their applications in logic.)

Topological algebra, the study of algebras with continuous operations, has itsorigins in classical mathematics, perhaps most notably in Galois theory: Galoisgroups of field extensions are profinite groups [77], and profinite algebras are animportant example of topological algebras (see §3.1.2). We will see that profinitelattices arise naturally as canonical extensions of lattices in finitely generatedvarieties of lattices.

The connections between canonical extensions and topological algebra aremany and intricate. The core connection, however, is that canonical extensionshave universal properties with respect to topological algebras. Meaning, that if wehave a lattice-based algebra A and an algebra homomorphism f : A→ B, whereB is a topological algebra, then there exists a unique continuous homomorphism

53

54 Chapter 3. Canonical extensions and topological algebra

f ′ : Aδ → B, extending f .

Aδ

f ′

A

eA>>~~~~~~~~

f// B

This is subject to certain assumptions on the algebras A and B, which we willreview in §3.4.

We begin this chapter with a review of preliminary facts about topologicalalgebras in §3.1. After that, we will first expand the canonical extension toolkitwith results that exploit topological lattice properties of canonical extensions oflattices in §3.2. We will then define lattice-based algebras and canonical extenionsof lattice-based algebras in §3.3. Finally in §3.4, we discuss the connection betweentopological lattice-based algebras and canonical extensions.

3.1 Topological algebra

In this section we will review certain useful facts about two important classes oftopological algebras, namely compact Hausdorff algebras and profinite algebras,which form a subclass of the compact Hausdorff algebras. After that we will take acloser look at topological lattices, which will lead to a characterization theorem forBoolean topological lattices. The idea behind topological algebra is very simple.Given an algebra signature Ω, an Ω-algebra is a structure A = 〈A, (ωA)ω∈Ω〉consisting of a set A and functions ωA : Aar(ω) → A for all ω ∈ Ω. A topological Ω-algebra is a structure 〈A, (ωA)ω∈Ω, τA〉 such that 〈A, τA〉 is a topological space andeach ωA : Aar(ω) → A is a (τ ar(ω), τ)-continuous function. This makes a topologicalalgebra A = 〈A, (ωA)ω∈ΩτA〉 into an object which has one foot in the world ofalgebra and one foot in the world of topology.

3.1.1. Example. If A = 〈A, (ωA)ω∈Ω〉 is an Ω-algebra, then 〈A, (ωA)ω∈Ω,P(A)〉is a topological algebra, where P(A) is the discrete topology on A. Although thisexample is rather trivial, we shall see when defining profinite algebras that it isvery important.

In §3.1.1, we will first briefly look at compact Hausdorff topological algebras,or compact Hausdorff algebras for short, which form the most general class oftopological algebras we will consider. We will look at the way the subalgebraconstruction behaves when it comes to compact Hausdorff algebras, and we willconsider the compactification functor for compact Hausdorff algebras. After that,we will review some facts concerning profinite algebras in §3.1.2. Profinite algebraswill play an important role later on in this chapter, so we will go into some detail tomake the reader familiar with their definition and construction. Finally, in §3.1.3we will look at a particular class of topological algebras, namely topological lattices.

3.1. Topological algebra 55

We will review some of the key facts about compact Hausdorff lattices, most notablythe fact (due to J.D. Lawson) that the topology on a compact Hausdorff lattice isunique. We will then specialize further by looking at Boolean topological lattices,i.e. lattices with a compact Hausdorff zero-dimensional topology and continuousmeet and join. What is nice about Boolean topological lattices is the fact thatone can characterize them order-theoretically, a result due to H.A. Priestley.

3.1.1 Compact Hausdorff algebras

In this subsection we will state several facts about topological algebras for whichthe topology is compact Hausdorff. One of the things which make compactHausdorff algebras particularly interesting is the fact every Ω-algebra A has aunique compactification ηA : A→ β A with the universal property that wheneverwe have a compact Hausdorff algebra 〈B, τ〉 and an algebra homomorphismf : A→ B, then there exists a unique continuous f ′ : β A→ B.

β Af ′

A

ηA>>||||||||

f// B

We like to think of β A as an extension of A, and of f ′ as a continuous extensionof f , however there is one important caveat to this perspective: in general, thenatural map ηA : A → β A is not an embedding. This is quite contrary to thenotion of compactification from general topology, where a compactification of aspace X is a compact space Y of which X is a dense subspace.

3.1.2. Fact ([84]). For any Ω-algebra A there exists a compact Hausdorff Ω-algebra 〈β A; τA〉 and a homomorphism ηA : A→ β A such that

1. ηA[A] is a dense subalgebra of 〈β A, τβ A〉,

2. for every compact Hausdorff algebra B and every f : A→ B, there exists aunique continuous homomorphism f ′ : β A→ B such that f ′ ηA = f .

β Af ′

A

ηA>>||||||||

f// B

3.1.3. Remark. Recall that a variety V is called residually small if there exist abound on the cardinality of the subdirectly irreducible algebras in V . It has beenconjectured [12, p. 519] that a sufficient condition for ηA : A→ β A being injectiveis that A ∈ V for some residually small variety V . However, at this time only theconverse of this conjecture is known to hold.


3.1.4. Fact (Corollary of Th. 1.2 of [85]). Let V be a variety. If ηA : A→UβA is an embedding for every A in V, then V is residually small.

3.1.5. Fact (Th. 2.1.9 of [33]). Let f, g : X → Y be continuous maps from atopological space X into a Hausdorff space Y . If there is a dense subset Z ⊆ Xsuch that f Z = g Z, then f = g.

The following Lemma is an elementary fact from topological algebra. Observethe distinction between the topological algebra 〈B, τ〉 and the discrete algebras A,B.

3.1.6. Lemma. Let 〈B, τ〉 be a Hausdorff topological algebra and let A be a subal-gebra of B such that A is dense in 〈B, τ〉. Then B ∈ HSP(A).

Proof We will show that if s ≈ t is an equation such that A s ≈ t, thenalso B s ≈ t. Suppose s and t use n variables. Consider the term functionssA : An → A and tA : An → A. Because A s ≈ t and A is a subalgebra of B,we know that sB and tB agree on An. Now because B is a topological algebra,the term functions sB : Bn → B and tB : Bn → B are continuous. But An is densein the Hausdorff space 〈Bn, τn〉, so by Fact 3.1.5, sB = tB. We conclude thatB s ≈ t.

3.1.7. Corollary. If A is an Ω-algebra, then β A ∈ HSP(A).

3.1.2 Profinite algebras and profinite completions

In this subsection, we will introduce two notions which are central to this chapter,namely profinite algebras and profinite completions. Profinite algebras are a verynatural and well-behaved example of topological algebras, which have been studiedextensively in the setting of groups [77] because of the connections between Galoistheoy and profinite groups.

To be able to say what profinite algebras and profinite completions are, wemust first introduce the notion of a limit of a diagram of algebras. The way onedefines a limit is from the bottom up, through a universal property with respectto a diagram of objects. Roughly, this means that every mapping going intothe diagram of algebras has to factor uniquely through the limit of the diagram.In this sense, a limit is a single object which represents a collection of objects,viz. the diagram of which it is a limit. At the same time, however, we can viewa limit from the top down as a single object which can be broken down into arepresentation in the form of the diagram of which it is a limit. This perspectiveis particularly salient in the case of profinite algebras, which are those algebraswhich arise as limits of diagrams of finite algebras. It is precisely the fact that aprofinite algebra A ' lim←−I Ai can be broken down into a diagram of finite algebrasAi that allows us to define a topology τ on A making 〈A, τ〉 a topological algebra.


A second important consequence of the fact that every profinite algebra Bhas a categorical representation as a diagram, is that it allows us to associatewith each algebra A a unique profinite algebra A, the profinite completion of Aand a natural map µA : A → A, such that whenever we have a homomorphismf : A→ B from A to a profinite algebra B, then f will factor through the naturalmap µA : A→ A:

Af ′

A

µA??

f// B

We will make frequent use of the universal property of µA : A→ A in §3.4.

Poset-indexed limits of Ω-algebras

We will now show how to define limits of diagrams of algebras, where we restrictourselves to the case in which the diagrams are indexed by a poset (see Remark3.1.11). This will provide us with the technical means to define profinite algebrasand profinite completions later on. Fix an algebra signature Ω and let AlgΩ

denote the category of Ω-algebras and Ω-algebra homomorphisms. Let 〈I,≤〉 bea poset. An I-indexed diagram in AlgΩ is a functor from I to AlgΩ. In otherwords, an I-indexed diagram is an assigmnent of an Ω-algebra Ai to each i ∈ I,and a homomorphism fij : Ai → Aj to each (i, j) ∈ I × I such that i ≤ j, withthe additional restrictions that

• for all i ∈ I, fii = idAi : Ai → Ai;

• if i ≤ j ≤ k, then fjk fij = fik.

We denote this diagram by 〈Ai, fij〉I . If the index poset I of a diagram 〈Ai, fij〉I ise.g. co-directed, we say that 〈Ai, fij〉I is a co-directed diagram of algebras. (Recallthat a poset I is co-directed if for all i, j ∈ I there exists k ∈ I such that k ≤ i, j.)

A cone to a diagram of algebras 〈Ai, fij〉I in AlgΩ is an I-indexed collectionof maps (gi : A→ Ai)I with a common domain A such that for all i, j ∈ I withi ≤ j, we have fij gi = gj.

Agi

gj

AAAAAAAA

Ai fij// Aj

Let (hi : B→ Ai)I be another cone to 〈Ai, fij〉I and let e : B→ A be an algebrahomomorphism. We say e is a map of cones if for all i ∈ I, gi e = hi.

Bhi

e // A

gi~~~~~~~

Ai


We call (gi : A → Ai)I a limiting cone if for all (hi : B → Ai)I , there exists aunique map of cones e : B → A. We then call A the limit of 〈Ai, fij〉I , writingA ' lim←−I Ai.

3.1.8. Example. If I is a discrete poset, meaning that i ≤ j iff i = j, then thelimit of an I-indexed diagram 〈Ai, fij〉I is simply the product: lim←−I Ai '

∏I Ai.

Indeed, if we have a collection of maps (B hi−→ Ai)I , then these define a uniquemap e : B→

∏I Ai, viz.

e : a 7→ (hi(a))i∈I .

Moreover, e is a map of cones: if a ∈ A and i ∈ I, then πi e(a) = hi(a).

Now that we have abstractly defined what limits are, we may ask ourselveswhether every diagram of algebras actually has a limit.

3.1.9. Fact. Every poset-indexed diagram 〈Ai, fij〉I in AlgΩ has a limiting cone,which may be computed as follows:

lim←−IAi = α ∈∏

IAi | ∀i, j ∈ I, i ≤ j ⇒ fij(α(i)) = α(j) .

Moreover, lim←−I Ai is a subalgebra of∏

I Ai.

Because varieties of algebras (§A.6) are closed under taking products and subalge-bras, we get as a corollary that varieties are closed under taking limits.

3.1.10. Corollary. If V is a variety of Ω-algebras, and if 〈Ai, fij〉I is a diagramof Ω-algebras such that Ai ∈ V for each i ∈ I, then also lim←−I Ai ∈ V.

We now know enough about limits of diagrams of algebras to define profinitealgebras and profinite completions.

3.1.11. Remark. What we have described here is really only a special case ofthe categorical notion of limit. The special case of poset-indexed diagrams willallow us to define what profinite algebras and profinite completions are. For adiscussion of the category theory at work here, see [69, §III.4].

Profinite Ω-algebras

We will now define profinite algebras and we will sketch why profinite algebrasare topological algebras. Let us start by simply giving the definition.

3.1.12. Definition. Let A be an Ω-algebra. We say that A is a profinite Ω-algebra if there exists a poset-indexed diagram 〈Ai, fij〉I such that

• A ' lim←−I Ai;


• for each i ∈ I, Ai is finite;

• the index poset I is co-directed, i.e. for all i, j ∈ I, there exists k ∈ I suchthat k ≤ i, j.

Let 〈Ai, fij〉I be a co-directed diagram of finite Ω-algebras and let (πi : A→ Ai)Ibe a limiting cone for this diagram, i.e. assume that A ' lim←−I Ai. We will nowsketch how we can use the limiting cone (πi : A→ Ai)I to define a topology onA, such that for each ω ∈ Ω, ωA is continuous. First observe that the followingcollection of sets forms a base for a topology on A:

π−1i (U) | U ⊆ Ai, i ∈ I

.

To see why, consider two basic opens π−1i (U) and π−1

j (V ), where U ⊆ Ai andV ⊆ Aj. We will show that there exists a k ∈ I and W ⊆ Ak such that

π−1i (U) ∩ π−1

j (V ) = π−1k (W ).

The key observation is the fact that by co-directedness of the index poset I,there must exist k ∈ I such that k ≤ i, j, so there exist maps fki : Ak → Ai andfkj : Ak → Aj. Because (πi : A→ Ai)I is a cone, it must be the case that

πi = fki πk and πj = fkj πk. (3.1)

We now see that

π−1i (U) ∩ π−1

j (V )

= (fki πk)−1(U) ∩ (fkj πk)−1(V ) by (3.1),

= π−1k f

−1ki (U) ∩ π−1

k f−1kj (V ) by basic set theory,

= π−1k

(f−1ki (U) ∩ f−1

kj (V ))

idem.

Thus we see that if we take W := f−1ki (U)∩f−1

kj (V ), then indeed π−1i (U)∩π−1

j (V ) =

π−1k (W ). We call the topology generated by π−1

i (U) | U ⊆ Ai, i ∈ I the profinitetopology on A. Note that the profinite topology on A is determined by the limitingcone (πi : A → Ai)I ; it is the coarsest topology which makes all the projectionmaps πi : A→ Ai continuous with respect to the discrete topology on Ai. We willnow sketch why A is a topological algebra with respect to the profinite topology.Consider an operation ω ∈ Ω and suppose for the sake of simplicity that ω isunary. We want to show that ωA is continuous. It suffices to show that for a basicopen set π−1

i (U), where U ⊆ Ai, it is also the case that ω−1A (π−1

i (U)) is open. Thekey observation is the fact that πi : A→ Ai is an Ω-algebra homomorphism; thisallows us to see that

ω−1A(π−1i (U)

)= (πi ωA)−1(U) by basic set theory,

= (ωAi πi)−1(U) because πi is an Ω-hom.,

= π−1i

(ω−1

Ai (U))

by basic set theory,


so that ω−1A (π−1

i (U)) is in fact a basic open.Thus, we see that if we are given a profinite algebra A ' lim←−I Ai, then A is a

topological algebra in its profinite topology, which we defined in a natural wayusing the limiting cone (πi : A→ Ai)I . We will now record the fundamental factthat every profinite topology is Boolean, i.e. every profinite topology is compact,Hausdorff and zero-dimensional.

3.1.13. Fact ([8]). If A ' lim←−Ai is a profinite algebra, then A is a Booleantopological algebra in its profinite topology.

3.1.14. Remark. We have chosen to first describe profinite algebras using uni-versal algebra, viz. as the limit of a diagram of finite algebras 〈Ai, fij〉I , and thento indicate that one can define a topology on lim←−I Ai, making lim←−I Ai a topologicalalgebra. Alternatively, one can start by endowing each algebra Ai with the discretetopology, and then show that one can also take limits of topological algebras, sothat it follows immediately that lim←−I Ai is a topological algebra. One can thenshow that lim←−I Ai is a closed subalgebra of

∏I Ai, so that it follows from general

topology that the profinite topology on lim←−I Ai is a Boolean topology.

Next, we would like to know how we can construct new profinite algebras fromone or more given profinite algebras. The most straightforward construction wecan use to create new profinite algebras is to take products.

3.1.15. Fact ([8]). If Ai | i ∈ I is a set of profinite algebras, then∏

I Ai isalso profinite.

(An alternative reference for the above fact is [77, Proposition 2.2.1].) Now whatabout subalgebras? That is, when is a subalgebra of a profinite algebra againprofinite? It turns out that here the profinite topology is very useful.

3.1.16. Fact ([8]). Let A be a profinite algebra. If B is subalgebra of A which isclosed in the profinite topology, then B is also profinite.

(An alternative reference for the above fact is [77, Corollary 1.1.8].) For otherconstructions, such as homomorphisms, things are more complicated. We willreturn to this matter in the special case of canonical extensions of profinite latticeswith Lemma 3.2.11.

Before we move on to profinite completions, we mention a result regardingan important question in the study of profinite algebras. We have seen above inFact 3.1.13 that every profinite Ω-algebra is a Boolean topological algebra in itsprofinite topology. Conversely, it well known that every Boolean space is profinite.This raises the following question: given a Boolean topological algebra 〈A, τ〉, canwe find a co-directed diagram of finite algebras 〈Ai, fij〉I such that A ' lim←−I Aiand τ is the profinite topology, i.e. the topology induced by (πi : A→ Ai〉I?


3.1.17. Fact ([24, 28]). Let 〈A, τ〉 be a Boolean topological algebra and supposethat A ∈ V for some variety V. If

1. V is finitely generated and congruence distributive, or

2. V has equationally definable principle congruences,

then there exists a co-directed diagram of finite algebras 〈Ai, fij〉I in V such thatA ' lim←−I Ai and τ is the profinite topology on A.

We will return to this question in §4.2.

Profinite completion of Ω-algebras

We now arrive at a very important property of profinite algebras, namely that everyalgebra A has a profinite completion, denoted A, and a natural map µA : A→ Awhich has the universal property that for every algebra homomorphism f : A→ Bto a profinite algebra B, there exists a unique continuous homomorphism f ′ : A→ Bsuch that f ′ µA = f .

Af ′

A

µA??

f// B

In our discussion of the profinite completion we will focus on the constructionof A and the natural map µA : A→ A, since we will use these later in this chapterwhen we describe the fundamental connection between canonical extensions andprofinite completions in §3.4.1.

Fix an Ω-algebra A; we will now describe how to construct A. We will do thisby specifying the diagram of which A is the limit. We define the following indexposet:

ΦA = θ ∈ ConA | A/θ is finite,where the order is the inclusion relation. Now if θ, ψ ∈ ΦA such that θ ⊆ ψ, thenthe Isomorphism Theorems of universal algebra [23, §II.6] tell us that the map

fθψ : a/θ 7→ a/ψ

is a well-defined, surjective Ω-algebra homomorphism. It is now easy to see that〈A/θ, fθψ〉ΦA forms a diagram in AlgΩ; we define

A := lim←−ΦAA/θ.

But while we are at it, we can immediately also define the natural map µA : A→ A.The quotient maps (µθ : A→ A/θ)ΦA , defined as

µθ : a 7→ a/θ,


form a cone to the diagram 〈A/θ, fθψ〉ΦA . Thus we can define µA : A→ A to be the

unique map of cones from A to A induced by (µθ : A→ A/θ)ΦA . We summarizethese two definitions below.

3.1.18. Definition. Let A be an Ω-algebra. We define A, the profinite comple-tion of A, to be the limit of the diagram of finite quotients of A.

A := lim←−ΦAA/θ

Additionally, we define µA : A→ A to be the map of cones induced by (µθ : A→A/θ)ΦA .

Now that we have defined the profinite completion, we record three importantfacts about it. The first is the universal property that we referred to before. Forgeneral categorical reasons, this property defines the profinite completion up toisomorphism.

3.1.19. Fact ([8]). Let f : A → B be an Ω-algebra homomorphism between Ω-algebras A and B. If B is profinite then there exists a unique continuous homo-morphism f ′ : A→ B such that f ′ µA = f .

Af ′

A

µA??

f// B

The second important fact is that strictly speaking, the name ‘profinite com-pletion’ can be seen as a misnomer. One would expect a completion of an algebraA to be an extension of A; however, the natural map µA : A→ A is not always anembedding. Recall that an Ω-algebra A is residually finite if for all a, b ∈ A witha 6= b, there exists a finite Ω-algebra B and an Ω-homomorphism f : A→ B suchthat f(a) 6= f(b).

3.1.20. Fact. An algebra A is residually finite iff the natural map µA : A→ Ais injective. A variety V is residually finite iff for every A ∈ V, µA : A → A isinjective.

In light of the above fact, we must make a distinction between a given algebraA and its image µA[A] in the statement of the fact below.

3.1.21. Fact. The subalgebra µA[A] is dense in A.


3.1.3 Topological lattices

We conclude this section with several observations about compact Hausdorf latticesand the characterization theorem for Boolean topological lattices.

Compact Hausdorff lattices and lattice-based algebras are an interesting classof topological algebras because a compact Hausdorff topology on a compactHausdorff lattice is always unique and intrinsic, meaning that the topology isuniquely determined by the algebra structure. This means that admitting acompact Hausdorff topology is a property of lattices. Moreover, by extension,admitting a compact Hausdorff topology is a property of lattice-based algebras.In this subsection we will present both known and new results which help us tounderstand topological properties of compact Hausdorff, Boolean topological andprofinite lattices via order theory and lattice theory. In particular, we will presentnecessary and sufficient conditions for a lattice to admit a unique Boolean topology(Theorem 3.1.26), and we will look at sufficient conditions for establishing thatsublattices of profinite lattices are again profinite.

Compact Hausdorff lattices

Let us start by recording the fundamental fact that the topology on a compactHausdorff lattice is intrinsic and unique.

3.1.22. Fact ([68]). Let L be a lattice. There exists at most one topology τ onL such 〈L, τ〉 is a compact Hausdorff topological lattice.

This fact has many important consequences, the first of which is that we maynow refer to a compact Hausdorff lattice 〈L, τ〉 simply by referring to L, sincethe topology τ is intrinsic. Recall that Lat is the category of bounded latticesand lattice homomorphisms, and that CLat is the category of complete latticesand complete lattice homomorphisms. By KHausLat we denote the category ofcompact Hausdorff topological lattices and continuous lattice homomorphisms;additionally, we are interested in BoolLat, which has Boolean topological latticesas its objects and continuous lattice homomorphisms as its morphisms, andin Pro- Latf , which is the category of profinite lattices and continuous latticehomomorphisms.

3.1.23. Fact. Let L be a compact Hausdorff lattice. Then

1. the unique topology making L a compact Hausdorff topological lattice is σ(L),the bi-Scott topology;

2. L is complete;

3. a lattice homomorphism f : L → M between compact Hausdorff lattices iscontinuous iff it is complete.


Consequently,

4. KHausLat, BoolLat and Pro- Latf are full subcategories of CLat.

For proofs of the statements in the above fact we refer the reader to [54, §VII-1.7]and [45, §VII-2]. We now shift our attention from compact Hausdorff lattices toBoolean topological lattices and profinite lattices.

Boolean topological lattices

We will now consider Boolean topological lattices, which form a subclass of compactHausdorff lattices. What is particularly nice about Boolean topological latticesis that one can characterize them order-theoretically, a result which was firstpublished by H.A. Priestley [76]. Below, we will discuss this result and its proofin some detail.

Since Boolean topological lattices are ordered Boolean topological spaces, wemay ask ourselves if they are perhaps Priestley spaces, that is if they satisfy thePriestley separation axiom. This is indeed the case.

3.1.24. Lemma. If L is a Boolean topological lattice, then 〈L, σ〉 is a Priestleyspace, i.e. for all x, y ∈ L such that x y there exists a clopen upper set U suchthat x ∈ U 63 y.

Proof Let x, y ∈ L with x y. Then by [54, Lemma VII-1.5] and [54, CorollaryVII-1.2], there exists an σ-open upper set U ′ ⊆ L such that x ∈ U ′ 63 y. Nowbecause we assumed that the σ-topology on L is zero-dimensional, there existssome clopen U ⊆ U ′ such that x ∈ U . Observe that y /∈ ↑U , since ↑U ⊆ U ′. Nowit follows by [5, Lemmas 2 and 3] that ↑U is clopen.

So now that we know that there exists an abundance of clopen upper sets in aBoolean topological lattice, we may ask ourselves what these sets look like. Recallthat given a lattice L, we denote the set of its compact elements by KL; p ∈ L iscompact if for all directed S ⊆ L such that

∨S exists, we have that p ≤

∨S only

if ↑ p G S.

3.1.25. Lemma. Let L be a compact Hausdorff lattice and let U, V ⊆ L be anupper and a lower set, respectively. Then

1. U is clopen iff there exists a finite Z ⊆ KL such that U = ↑Z;

2. V is clopen iff there exists a finite W ⊆ K(Lop) such that V = ↓W .

Proof We will only prove (1), since (2) is just its order dual. Suppose that U ⊆ Lis a clopen upper set. We denote the minimal elements of U by minU :

minU := p ∈ U | ∀x ∈ L, x < p⇒ x /∈ U.


Now we claim thatU = ↑minU. (3.2)

To see why, suppose that x ∈ U . Then ↓x is σ↑-closed and hence, σ-closed;consequently U ∩ ↓x is σ-closed. Now by Lemma 2.1.16, U ∩ ↓ x is closed underco-directed meets, so a fortiori it is closed under taking meets of chains. In otherwords, U ∩↓x is a partial order which is closed under meets of chains; it follows by(the order dual of) Zorn’s Lemma that there exists a minimal element p ∈ U ∩ ↓x.It is easy to see that p is also minimal in U ; it follows that (3.2) holds. Next weclaim that

minU ⊆ KL. (3.3)

To see why, consider p ∈ minU and suppose that p ≤∨S for some directed S ⊆ L;

we must show that there is an x ∈ S such that p ≤ x. Because L is a compactHausdorff lattice, we know that ∧ : L × L → L is (σ, σ)-continuous; by Lemma2.1.17, it follows that ∧ is (σ↑, σ↑)-continuous, i.e. that ∧ preserves directed joins.So since p ≤

∨S, we see see that

p = p ∧∨S =

∨p ∧ S.

Now suppose towards a contradiction that p ∧ x < p for all x ∈ S; then since pis minimal in U , it follows that p ∧ S ⊆ L \ U . Because U is open, it follows byLemma 2.1.16 that

∨p ∧ S ∈ L \ U . But this contradicts the fact that∨

p ∧ S = p ∈ U.

We conclude that there must be some x ∈ S such that p ≤ x. It follows thatp ∈ KL. Now recall that by (3.2),

U = ↑minU =⋃p∈U ↑ p.

Since U is σ-closed and each ↑ p is σ-open by (3.3) and Fact A.5.5(2), it followsby compactness that there must exist some finite Z ⊆ minU ⊆ KL such thatU = ↑Z.

Conversely, suppose that U = ↑Z for some finite Z ⊆ KL. Since

U = ↑Z =⋃p∈Z ↑ p,

we see firstly that U is open by Fact A.5.5(2). Secondly, we see that U is closedbecause it is a finite union of σ↓-closed sets, and σ↓ ⊆ σ. It follows that U isclopen.

A lattice L is called bi-algebraic if both L and Lop are algebraic. By λ(L) :=σ↑(L) ∨ ι↓ we denote the Lawson topology of L [45].

We are now ready to prove the main result of this section, which is theCharacterization Theorem for Boolean topological lattices, due to H.A. Priestley.Observe however that condition (2) below is new.


3.1.26. Theorem ([76]). Let L be a lattice. The following are equivalent:

1. L is a Boolean topological lattice;

2. L is complete and there exist P,Q ⊆ L such that

(a) P is join-dense in L and Q is meet-dense in L;

(b) for every p ∈ P , there is a finite Z ⊆ L such that L \ ↑ p = ↓Z;

(c) for every q ∈ Q, there is a finite Z ⊆ L such that L \ ↓ p = ↑Z;

3. L is complete and bi-algebraic, σ↑ = ι↑ and σ↓ = ι↓.

Proof (1) ⇒ (2). We will first argue that L is bi-algebraic. To see that L isalgebraic, it suffices to show that if x, y ∈ L and x y, then there exists a p ∈ KLsuch that p ≤ x and p y. But this is easy to see: by Lemma 3.1.24, there existsa clopen upper set U ⊆ L such that x ∈ U 63 y. By Lemma 3.1.25, U = ↑Z forsome finite Z ⊆ KL. But then it follows that there must be some p ∈ Z such thatp ≤ x; moreover, since y /∈ ↑Z, it follows that p y. The argument for showingthat L is co-algebraic is identical. If we now define P := KL and Q := KLopthen it follows that (2)(a) holds. To see that (2)(b) holds, take any p ∈ P = KL.It follows by Lemma 3.1.25 that ↑ p is a clopen upper set, so L \ ↑ p must be aclopen lower set. Applying Lemma 3.1.25 again, we see that there must be someW ⊆ KLop such that L \ ↑ p = ↓W . The proof for (2)(c) is order dual.

(2) ⇒ (3). We will first show that

σ↑ = ι↑ and σ↓ = ι↓. (3.4)

We will only show the first part of (3.4), since the other follows by order duality.Recall from Lemma 2.1.17(1) that ι↑ ⊆ σ↑, so it suffices to show that σ↑ ⊆ ι↑.Now observe that by assumption (2)(b), we know that for every p ∈ P , we have

L \ ↑ p = ↓Z =⋃q∈Z ↓ q,

for some finite set Z, so that

↑ p = L \ (L \ ↑ p) = L \(⋃

q∈Z ↓ q)

=⋂q∈Z(L \ ↓ q).

We see that ↑ p is a finite intersection of ι↑-open sets; since p ∈ P was arbitrary,we see that

for all p ∈ P , ↑ p is ι↑-open. (3.5)

Now let U be a σ↑-open set and let x ∈ U . Because P is join-dense in L, we knowthat x =

∨(↓x∩P ). Since U is σ↑-open, it follows that there must be some finite

Z ⊆ ↓x ∩ P such that∨Z ∈ U . Now observe that

x ∈ ↑∨Z since Z ⊆ ↓x ∩ P ,

=⋂p∈Z ↑ z by order theory,

⊆ U since∨Z ∈ U .


Since⋂p∈Z ↑ z is ι↑-open by (3.5) and since x ∈ U was arbitrary, it follows that

U is ι↑-open. Since U was an arbitrary σ↑-open set it follows that σ↑ ⊆ ι↑ andconsequently, (3.4) holds. Now we will show that

L is bi-algebraic. (3.6)

We first show that P ⊆ KL. If p ∈ P , then by (3.5) and (3.4), ↑ p is σ↑-open.But then it follows immediately that p ∈ KL: if p ≤

∨S for some directed S ⊆ L,

then∨S ∈ ↑ p, so there exists some y ∈ S such that p ≤ y. Now we see that for

any x ∈ L,

x =∨

(↓x ∩ P ) because P is join-dense in L,

=∨

(↓x ∩KL) by order theory because P ⊆ KL,

so that we see that L is algebraic. By an order dual argument it follows that L isalso co-algebraic, so that (3.6) holds.

(3) ⇒ (1). We will first show that the σ-topology on L is a Boolean topology,i.e. it is compact, Hausdorff and zero-dimensional. Recall that the Lawson topologyon L is defined as λ = σ↑∨ ι↓. Since σ := σ↑∨σ↓, it follows by our assumption thatin our case, λ = σ. Since L is algebraic by assumption, it follows by [45, TheoremIII.1.10] that the Lawson topology on L is compact and Hausdorff; consequently,so is the σ-topology. Now if x, y ∈ L such that x y, then since L is algebraic,there exists a p ∈ KL such that p ≤ x and p y, i.e. x ∈ ↑ p 63 y. Becausep ∈ KL, ↑ p is σ↑-open. Because ↑ p is a principal upper set, it is ι↓-closed. Byour assumption that σ↑ = ι↑ and σ↓ = ι↓, it follows that ↑ p is σ-clopen, so sincex, y ∈ L were arbitrary, it follows that the σ-topology is totally disconnected;consequently, the σ-topology is a Boolean topology.

Finally, we will show that L is a topological lattice in its σ-topology. We knowby associativity that ∧ : L × L → L preserves co-directed meets, so that ∧ is(σ↓, σ↓)-continuous. Because L is algebraic, we know by [45, Proposition I-1.14]that ∧ preserves directed joins, i.e. that ∧ is (σ↑, σ↑)-continuous. It follows that ∧is (σ, σ)-continuous; the argument for ∨ : L× L→ L is order-dual. We concludethat L is a Boolean topological lattice.

3.1.27. Corollary. If L is a Boolean topological lattice, then σ = λ, i.e. itsintrinsic Boolean topology is the Lawson topology.

We conclude this section with an application of the Characterization Theoremabove. Recall that we saw above that a closed sublattice of a profinite lattice isagain profinite. Using Theorem 3.1.26 in conjunction with a technical result fromdomain theory, we can prove the following:

3.1.28. Lemma. Let L be a profinite lattice. If L′ is a complete subalgebra of Lthen L′ is closed and hence, profinite.


Proof Because L is algebraic by Theorem 3.1.26, it is a continuous lattice in thesense of [45]. Consequently, we may apply [45, Theorem III.1.11] to conclude thatL′ must be closed in the Lawson topology of L. Now by Corollary 3.1.27, we knowthat σ = λ, so L′ is a σ-closed subalgebra of L. Since the profinite topology on Lmust be the σ-topology (as there can be only one compact Hausdorff topologyon L which is compatible with ∧ and ∨), we see that L′ is closed in the profinitetopology on L. It now follows from Lemma 3.1.16 that L′ is itself profinite.

This concludes our preliminaries for this chapter on topological algebra andtopological lattices.

3.2 Canonical extensions of maps II: maps into

profinite lattices

In §2.2, we defined the canonical extension of an order-preserving map f : L→Mby first extending f to a map F f : F L → FM, mapping a filter F to ↑ f [F ],and subsequently extending F f to a continuous extension fO : Lδ → Mδ, anddually via I L when constructing fM. This approach fails, however, if we drop theassumption that f is order-preserving, because F f is then no longer defined. Inthis section we will introduce a different way of continuously extending f : L→Mto a map fO : Lδ →Mδ, which does not depend on any properties of f . We canstill characterize fO as a largest continuous extension of f if we make additionalassumptions about M, the codomain of f .

Some of the definitions and results in this section are generalizations from thecase of distributive lattices studied by Gehrke and Jonsson [39]. As we indicatedin our paper with M. Gehrke [43] however, distributivity of the underlying latticeis not an essential property for ensuring good behaviour of the canonical extension.Rather, what matters is that the underlying lattice lies in a finitely generatedvariety. We will make use of the technical lemmas from this section in the rest ofthis chapter, when we look at canonical extensions of lattice-based algebras ratherthan plain lattices.

The section is organized as follows. First, we show that there is a naturalway to define a lower and upper extension of an arbitrary map f : L→ C, from alattice into a complete lattice, to a continuous map f ′ : Lδ → C, viz. the lim infand lim sup extension of f . We then use lim inf and lim sup to extend an abitraryfunction f : L→M to a continuous function fO : Lδ →Mδ in §3.2.1. We will thenshow in §3.2.2 how we can view fO : Lδ →Mδ as a maximal continuous extensionof f : L→M if we assume that HSPM is finitely generated. Finally, in §3.2.3 weinvestigate properties of extension of compositions of arbitrary maps. We concludethis section with an overview of the contributions and further work in §3.2.4.

3.2. Canonical extensions of maps II: maps into profinite lattices 69

3.2.1 Extending maps via lim inf and lim sup

Our first goal in this subsection is to extend an arbitrary map f : L→ C from alattice L to a complete lattice C to a continuous map f ′ : Lδ → C in such a waythat f ′ e = f , where e : L→ Lδ is the canonical extension of L.

Lδf ′ // C

L

e

OO

f

88qqqqqqqqqqqqq

We exploit the fact that e[L] is dense in 〈L, δ(L)〉 (by Lemma 2.1.28), meaningthat for ‘a lot’ of points in Lδ, our function f ′ : Lδ → C is already defined. Foran arbitrary x ∈ Lδ and a δ-open neighborhood U of x, we get a set of valuesf [e−1(U)] in C approximating f ′(x). We can now take the infimum (meet) orsupremum (join) of this set of approximating values. Thus, we get an inf- orsup-approximant of f ′(x) for every open neighborhood of x. Intuitively, we cannow define f ′(x) to be the ‘limit’ over all open neighborhoods of x of theseapproximants.

Recall that the δ-topology on Lδ has as its base the collection of sets

↑ eF(F ) ∩ ↓ eI(I) | F ∈ F L, I ∈ I L.

We now arrive at the following definition:

3.2.1. Definition. Given a function f : L→ C, where C is a complete lattice,we define lim inf f : Lδ → C, where

lim inf f(x) =∨∧

f [F ∩ I] | eF(F ) ≤ x ≤ eI(I).

Dually, we define lim sup f : Lδ → C as

lim sup f(x) =∧∨

f [F ∩ I] | eF(F ) ≤ x ≤ eI(I).

First, observe that these definitions follow the pattern we sketched above. Thereader may notice however that the expressions f [F ∩ I] are a lot simpler thanthe f [e−1(U)] we arrived at before. The reason lies in the following lemma:

3.2.2. Lemma. Let e : L→ Lδ be a canonical extension and let F ∈ F L, I ∈ I L.Then eF(F ) ≤ e(a) ≤ eI(I) iff a ∈ F ∩ I.

Proof We show that eF(F ) ≤ e(a) iff a ∈ F ; the statement then follows by orderduality. Now, observe that eF(F ) ≤ e(a) = eF(↑ a) iff F ⊇ ↑ a, since eF(F ) is anembedding (by Lemma 2.1.28). But F ⊇ ↑ a iff a ∈ F .


The following lemma shows that lim inf f indeed gives us a continuous extensionof f .

3.2.3. Lemma. Let f : L→ C be a function from a lattice L to a complete latticeC and let e : L→ Lδ be the canonical extension of L. Then

1. lim inf f is (δ, σ↑)-continuous;

2. lim sup f is (δ, σ↓)-continuous;

3. lim inf f ≤ lim sup f ;

4. lim inf f e = lim sup f e = f .

Lδlim inf f // C

L

e

OO

f

88qqqqqqqqqqqqq

Proof (1). First, observe that given x ∈ Lδ,∧f [F ∩ I] | eF(F ) ≤ x ≤ eI(I)

is a directed set: take F1, F2 ∈ F L and I1, I2 ∈ I L such that eF(Fi) ≤ x ≤ eI(Ii)for i = 1, 2. Then

eF(F1 ∩ F2) = eF(F1) ∨ eF(F2) since eF is a homomorphism,

≤ x by assumption,

≤ eI(I1) ∧ eI(I2) idem,

= eI(I1 ∩ I2) because eI is a homomorphism.

Moreover, since F1 ∩ F2 ∩ I1 ∩ I2 ⊆ Fi ∩ Ii for i = 1, 2, we get∧f [F1 ∩ F2 ∩ I1 ∩ I2] ≥

∧f [Fi ∩ Ii] for i = 1, 2;

it follows that lim inf f(x) is a directed join. We will use this fact to show thatlim inf f is locally continuous at x; since x is arbitrary this suffices to show thatlim inf f is continuous. Suppose that U ⊆ C is Scott-open and lim inf f(x) ∈ U ,then since lim inf f(x) is a directed join, there must be some F ′ ∈ F L, ′I ∈ I Lsuch that eF(F ′) ≤ x ≤ eI(I ′) and

∧f [F ′ ∩ I ′] ∈ U . But then for all y ∈ Lδ such

that eF(F ′) ≤ y ≤ eI(I ′), we have

lim inf f(y) =∨∧

f [F ∩ I] | eF(F ) ≤ y ≤ eI(I)

by definition of lim inf,

≥∧f [F ′ ∩ I ′] since eF(F ′) ≤ y ≤ eI(I ′),

∈ U,


so that it follows that lim inf f is locally (δ, σ↑)-continuous. Part (2) is just theorder dual of (1).

(3). Let x ∈ Lδ; we want to show that∨∧f [F ∩ I] | eF(F ) ≤ x ≤ eI(I)

≤∧∨

f [F ∩ I] | eF(F ) ≤ x ≤ eI(I).

It suffices to show that for all F, F ′ ∈ F L and I, I ′ ∈ I L such that eF(F ) ≤x ≤ eI(I) and eF(F ′) ≤ x ≤ eI(I ′), we have that

∧f [F ∩ I] ≤

∨f [F ′ ∩ I ′]. First,

observe that we have that

eF(F ∩ F ′) = eF(F ) ∨ eF(F ′) ≤ x ≤ eI(I) ∧ eI(I ′) = eI(I ∩ I ′),

as before. It follows by the compactness property of canonical extensions that(F ∩ F ′) G (I ∩ I ′), i.e. F ∩ F ′ ∩ I ∩ I ′ 6= ∅. But then∧

f [F ∩ I] ≤∧f [F ∩ F ′ ∩ I ∩ I ′] since (F ∩ I) ⊇ (F ∩ F ′ ∩ I ∩ I ′),

≤∨f [F ∩ F ′ ∩ I ∩ I ′] since F ∩ F ′ ∩ I ∩ I ′ 6= ∅,

≤∨f [F ′ ∩ I ′] since (F ∩ F ′ ∩ I ∩ I ′) ⊆ (F ′ ∩ I ′).

Since F, F ′ ∈ F L and I, I ′ ∈ I L were arbitrary it now follows that lim inf f(x) ≤lim sup f(x).

(4). We only show that lim inf f e = f ; the other equality follows by orderduality. Let a ∈ L, then

lim inf f e(a) =∨∧

f [F ∩ I] | eF(F ) ≤ e(a) ≤ eI(I)

by definition,

=∨∧f [F ∩ I] | a ∈ F ∩ I by Lemma 3.2.2,

≤ f(a) by order theory.

Conversely, since eF(↑ a) ≤ e(a) ≤ eI(↓ a), we see that

f(a) =∧f [↑ a ∩ ↓ a] ≤ lim inf f e(a).

3.2.4. Lemma. Let f : L1 → C1 and g : L2 → C2 be functions from lattices L1,L2 to complete lattices C1, C2. Then

lim inf(f × g) = (lim inf f)× (lim inf g)

lim sup(f × g) = (lim sup f)× (lim sup g)

Proof We will only prove the first statement, since the proof of the secondstatement is identical modulo order duality. Although we will have to do somebookkeeping and there is a lot of notation, the proof of this lemma is essentially veryeasy. We first make a number of observations. For starters, if (x1, x2) ∈ Lδ1 × Lδ2,then

(F, I) | eFL1×L2(F ) ≤ (x1, x2) ≤ eIL1×L2

(I)

=

(F1 × F2, I1 × I2) | eFLi(Fi) ≤ xi ≤ eILi(Ii), i = 1, 2, (3.7)


where F and I are filters and ideals of L1×L2, and Fi and Ii are filters and idealsof Li for i = 1, 2. This follows from fact that every filter of L1 × L2 is of the formF1 × F2 for some F1 ∈ F L1, F2 ∈ F L2. Next, we claim that for all filters andideals F1, I1 ⊆ L1 and F2, I2 ⊆ L2,

(F1 × F2) ∩ (I1 × I2) = (F1 ∩ I1)× (F2 ∩ I2). (3.8)

This follows from basic set theory. Finally, we claim that for all filters and idealsF1, I1 ⊆ L1 and F2, I2 ⊆ L2,

f × g [(F1 × F2) ∩ (I1 × I2)] = f [(F1 ∩ I1)]× g [(F2 ∩ I2)] . (3.9)

This follows from (3.8) and basic set theory, since f × g(a, b) = (f(a), g(b)) for(a, b) ∈ L1×L2. Let (x1, x2) ∈ Lδ1×Lδ2; we will show that lim inf(f × g)(x1, x2) =(lim inf f(x1), lim inf g(x2)).

lim inf f × g(x1, y1)

=∨∧

f × g[F ∩ I] | eFL1×L2(F ) ≤ (x, y) ≤ eIL1×L2

(I)

by definition of lim inf,

=∨∧

f × g[F1 × F2 ∩ I1 × I2] | eFLi(Fi) ≤ xi ≤ eILi(Ii), i = 1, 2

by (3.7),

=∨∧

f [(F1 ∩ I1)]× g [(F2 ∩ I2)] | eFLi(Fi) ≤ xi ≤ eILi(Ii), i = 1, 2

by (3.9),

=∨

(∧f [(F1 ∩ I1)] ,

∧g [(F2 ∩ I2)]) | eFLi(Fi) ≤ xi ≤ eILi(Ii), i = 1, 2

because

∧is computed component-wise,

=(∨∧

f [(F1 ∩ I1)] | eFL1(F1) ≤ x1 ≤ eIL1

(I1),∨∧

g [(F2 ∩ I2)] | eFL2(F2) ≤ x2 ≤ eIL2

(I2) )

because∨

is computed component-wise,

= (lim inf f(x1), lim inf g(x2))

by definition of lim inf.

Because (x1, x2) ∈ Lδ1 × Lδ2 was arbitrary, it follows that lim inf(f × g) =(lim inf f)× (lim inf g).

3.2.5. Remark. The technical results about lim inf may be seen even more astopological results rather than canonical extension results. This viewpoint isdiscussed further in [39, §2.3].

Thus we see that the lim inf-construction we have defined above has the desirableproperty that it gives us a continuous extension of an arbitrary map into a complete


lattice. We now have a good candidate definition for extending an arbitrary mapf : L→M to a map from Lδ toMδ: simply take the map lim inf(eMf) : Lδ →Mδ.

Lδlim inf(eMf) //Mδ

L

eL

OO

f//M

eM

OO

As the above diagram illustrates, lim inf(eMf) is indeed an extension of f : L→Mto a map from Lδ to Mδ. But before we take this as the definition, we must askourselves an important question: is this proposed extension compatible with theextensions of order-preserving maps we studied in §2.2?

3.2.6. Lemma. If f : L→M is order-preserving, then fO = lim inf(eM f) andfM = lim sup(eM f).

Proof Let x ∈ Lδ. Recall that

fO(x) =∨

eFMF f(F ) | eFL (F ) ≤ x,

and that

lim inf(eM f)(x) =∨∧

eM f [F ∩ I] | eFL (F ) ≤ x ≤ eIL(I).

We claim that

eFMF f(F ) =∧eM f [F ∩ I] whenever eFM(F ) ≤ eIM(I). (3.10)

First, observe that

eFMF f(F ) =∧eM [↑ f [F ]] by definition,

=∧eM f [F ] since eM is order-preserving.

Moreover, we see that∧eM f [F ∩ I] =

∧eM f [↑(F ∩ I)] since f is order-preserving,

=∧eM f [F ] by Lemma 2.1.3,

where we need the fact that eFM(F ) ≤ eIM(I) in order to apply Lemma 2.1.3. Itfollows that (3.10) holds; it is now easy to see that fO(x) = lim inf(eM f)(x).

In light of Lemma 3.2.6, we can now safely state the following definition, whichsubsumes Definition 2.2.1.


3.2.7. Definition. If f : L→M is an abitrary function between lattices, thenwe define fO : Lδ →Mδ as follows:

fO := lim inf(eM f).

Dually, we define fM : Lδ →Mδ:

fM := lim sup(eM f).

As a corollary of Lemma 3.2.3, we can now list the following properties of ournewly defined extensions fO and fM.

3.2.8. Corollary. Let f : L→ M be an arbitrary function between lattices L,M. Then

1. fO : Lδ →Mδ is (δ, σ↑)-continuous;

2. fM : Lδ →Mδ is (δ, σ↓)-continuous;

3. fO ≤ fM.

Proof This follows immediately from Lemma 3.2.3 and Definition 3.2.7.

3.2.2 Maps into profinite lattices

Recall from §2.2 that if f : L→M is an order-preserving map, then fO : Lδ →Mδ

is the largest (δ↑, ι↑)-continuous extension of f , where ι↑ is the upper intervaltopology. We would like to prove a similar result in the case that f is notnecessarily order-preserving. We begin with a maximality result concerning thelim inf-construction for maps f : L→ C, where we use the additional assumptionthat C is profinite, which is a very strong property. It is an interesting and openquestion whether the assumption that C is profinite is essential; see Remark 3.2.22.

3.2.9. Theorem. Let f : L → C be a function from a lattice L to a profinitelattice C, and let f ′ : Lδ → C be an extension of f , i.e. assume that f ′ eL = f .

Lδ

f ′

L

eL??~~~~~~~

f// C

1. If f ′ : Lδ → C is (δ, ι↑)-continuous, then f ′ ≤ lim inf f .

2. If f ′ : Lδ → C is (δ, ι↓)-continuous, then lim sup f ≤ f ′.


Proof By order duality it suffices to prove part (1). We will prove somethingstronger, in fact: we will show that for all x ∈ Lδ, if f ′ : Lδ → C is locally (δ, ι↑)-continuous at x, then f ′(x) ≤ lim inf f(x). Towards a contradiction, suppose thatf ′(x) lim inf f(x). By Theorem 3.1.26, C is algebraic, so there must exist acompact element p ∈ KC such that p ≤ f ′(x) and p lim inf f(x). Now ↑ p isσ↑-open (Fact A.5.5), so again by Theorem 3.1.26, it follows that ↑ p is ι↑-open.By local continuity of f ′ at x, there must exist F ∈ F L and I ∈ I L such thateFL (F ) ≤ x ≤ eIL(I) and

f ′[y ∈ Lδ | eFL (F ) ≤ y ≤ eIL(I)

]⊆ ↑ p. (3.11)

Now for every a ∈ F ∩ I, by Lemma 3.2.2 we have eFL (F ) ≤ eL(a) ≤ eIL(I), soby (3.11), f ′ eL(a) = f(a) ∈ ↑ p. Since a ∈ F ∩ I was arbitrary, it follows thatf [F ∩ I] ⊆ ↑ p, hence

∧f [F ∩ I] ≥ p. Since eFL (F ) ≤ x ≤ eIL(I), it follows by

definition of lim inf that

lim inf f(x) ≥∧f [F ∩ I] ≥ p.

But this contradicts our assumption that p lim inf f(x). It follows that indeed,f ′(x) lim inf f(x).

3.2.10. Corollary. Let f : L→ C be a function from a lattice L to a profinitelattice C. If f ′ : Lδ → C is a (σ, σ)-continuous function such that f ′ eL = f , wehave f ′ = lim inf f = lim sup f .

Proof If f ′ is (σ, σ)-continuous, then by Lemma 2.1.28(3), f ′ is also (δ, σ)-continuous. This has two immediate consequences. Firstly, since σ↑ ⊆ σ bydefinition, we see that f ′ is (δ, σ↑)-continuous, so by Theorem 3.2.9, f ′ ≤ lim inf f .Secondly, since σ↓ ⊆ σ, we get that f ′ is (δ, σ↓)-continuous, so it follows again byTheorem 3.2.9 that lim sup f ≤ f ′. We conclude that

lim sup f ≤ f ′ ≤ lim inf f.

Since lim inf f ≤ lim sup f by Lemma 3.2.3(3), it follows that lim sup f = f ′ =lim inf f .

With the help of the above theorem, we can make sure that fO : Lδ → Mδ

is the largest continuous extension of f : L→ M if Mδ happens to be profinite.Fortunately, there is a condition on M that guarantees that this will be the case.

3.2.11. Lemma. Let L be a profinite lattice and let M be an arbitrary lattice. Ifthere exists a complete surjective homomorphism h : L→Mδ then Mδ is a Booleantopological lattice.


Proof We know from Fact 2.1.30 that Mδ is join-generated by its completelyjoin-irreducibles J∞(Mδ) and meet-generated by its completely meet-irreduciblesM∞(Mδ). We will show that

∀p ∈ J∞(Mδ),∃Z ⊆Mδ finite, such that Mδ \ ↑ p = ↓Z, and (3.12)

∀p ∈ M∞(Mδ),∃Z ⊆Mδ finite, such that Mδ \ ↓ p = ↑Z.

Since h : L→Mδ is a complete homomorphism, it has a left adjoint h[ : Mδ → L.Since h[ a h and h is surjective, we know from Fact A.3.3 that h h[ = idMδ . Weclaim that

h[ maps elements of J∞(Mδ) to elements of KL. (3.13)

Let p ∈ J∞(Mδ); we will show that h[(p) ∈ KL. Let S ⊆ L be a directed set suchthat h[(p) ≤

∨S, then

h[(p) = h[(p) ∧∨S by order theory,

=∨(

h[(p) ∧ S)

by σ-continuity of ∧.

Now we see that

p = h h[(p) because h h[ = idMδ ,

= h(∨

(h[(p) ∧ S))

because h[(p) =∨

(h[(p) ∧ S),

=∨(

h h[(p) ∧ h[S])

because h is a complete lattice hom.,

=∨

(p ∧ h[S]) because h h[ = idMδ .

Since p is completely join-irreducible, it follows that there must exist x ∈ S suchthat p = p ∧ h(x). But then p ≤ h(x), so since h[ a h, we see that h[(p) ≤ x; itfollows that p ∈ KL and we may conclude that (3.13) holds.

Now if p ∈ J∞(Mδ), so that h[(p) ∈ KL, we know by Theorem 3.1.26 thatthere exists a finite Z ⊆ L such that L \ ↑h[(p) = ↓Z. We will show that

Mδ \ ↑ p = ↓h[Z]. (3.14)

Recall from Fact A.3.3 that h[ is an order embedding because h is surjective; wenow see that

x ∈ ↓h[Z]

iff ∃q ∈ Z, x ≤ h(q) by def. of ↓ ·,iff ∃q ∈ Z, h[(x) ≤ q because h[ a h,

iff h[(x) ∈ ↓Z by def. of ↓ ·,iff h[(p) h[(x) because L \ ↑h[(p) = ↓Z,

iff p x because h[ is an order embedding.


It follows that (3.14) holds. Now because Z is finite, so is h[Z]. It follows that(3.12) holds (the case for q ∈ M∞(Mδ) follows by order duality). Now we mayapply Theorem 3.1.26 to conclude that Mδ is a Boolean topological lattice.

3.2.12. Theorem. Let L be a lattice such that HSP(L) is finitely generated.Then Lδ is profinite and Lδ ∈ HSP(L)

Proof Since HSP(L) is finitely generated, there must exist some finite latticeM such that HSP(L) = HSP(M). Since M is finite, by Fact A.6.3 we have thatHSP(M) = HSPB(M). Since M clearly is profinite and has the property thatM ∈ HSP(M) = HSP(L), it suffices to show that this property is preserved as weapply PB, S and H.

If L ∈ PB(M), then there exists a Boolean decomposition (px : L→M)x∈X forsome Boolean space X. By Fact 2.1.27, we have that Lδ 'MX . It follows by Fact3.1.15 that Lδ is profinite; it is also immediate that Lδ ∈ HSP(L).

If L ∈ SPB(M), then there exists some L′ ∈ PB(M) such that L is a subalgebraof L′. It follows from Theorem 2.2.24 that Lδ is (isomorphic to) a complete subal-gebra of L′δ. By the above, L′δ is profinite and L′δ ∈ HSP(L), so it immediatelyfollows that Lδ ∈ HSP(L). Moreover, it follows by Lemma 3.1.28 that Lδ isprofinite.

Finally, if L ∈ HSPB(M), then there exists L′ ∈ SPB(M) and a surjectivehomomorphism h : L′ → L. By Theorem 2.2.24, hδ : L′δ → Lδ is a completesurjective homomorphism; it follows immediately that Lδ ∈ HSP(M) = HSP(L).Since L′δ is profinite by the above, it follows from Lemma 3.2.11 that Lδ is aBoolean topological lattice. Since Lδ ∈ HSP(M), which is a finitely generatedcongruence distributive variety, it follows from Fact 3.1.17 that Lδ is profinite.

3.2.13. Remark. In fact, we will later see in §3.4.2 the statement that Lδ isprofinite is equivalent to saying that Lδ is the profinite completion of L. In thislight, the above theorem is a consequence of the main result in [50].

3.2.14. Remark. In our proof of Theorem 3.2.12, we invoke Fact 3.1.17 to showthat under the assumptions of the theorem, a Boolean topological quotient of aprofinite lattice must again be profinite. In a recent paper, Gehrke et al. showthat a Boolean topological quotient of a profinite algebra is always profinite, usinga duality argument [35].

We can now state a powerful result about canonical extensions of arbitrary maps,which echoes Theorem 2.2.4.

3.2.15. Corollary. Let f : L→M be an arbitrary function between lattices Land M; furthermore assume that HSP(M) is finitely generated. Let f ′ : Lδ →Mδ

be an extension of f , i.e. assume that f ′ eL = eM f . Then


1. if f ′ : Lδ →Mδ is (δ, ι↑)-continuous, then f ′ ≤ fO;

2. if f ′ : Lδ →Mδ is (δ, ι↓)-continuous, then fM ≤ f ′;

Proof We only prove (1). If HSP(M) is finitely generated, then by Theorem3.2.12, Mδ is profinite and consequently eM f : L→Mδ is a map into a profinitelattice. It now follows by Theorem 3.2.9 that f ′ ≤ lim inf(eM f) = fO.

3.2.16. Corollary. Let f : L→M be an arbitrary function between lattices Land M; furthermore assume that HSP(M) is finitely generated. If f ′ : Lδ →Mδ isa (σ, σ)-continuous function such that f ′ eL = eM f then f ′ = fO = fM, i.e. fis smooth.

3.2.3 Canonical extension and function composition

We conclude this section with a series of technical lemmas about the interactionbetween canonical extension and function composition, and composition withlattice homomorphisms in particular.

If we look at a function composition with a lattice homomorphism on the left,we need no other assumptions to show that canonical extension commutes withfunction composition. Recall that if h : L→M is a lattice homomorphism then his smooth (by Theorem 2.2.18), so we write hδ instead of hO or hM.

3.2.17. Lemma. Let ei : Li → Lδi be canonical extensions of lattices L1,L2,L3; letf : L1 → L2 be an arbitrary map and let h : L2 → L3 be a lattice homomorphism.Then hδfO = (hf)O.

Proof Recall from Theorem 2.2.24 that hδ : Lδ2 → Lδ3 is a complete homomor-phism.

Lδ1fO// Lδ2

hδ // Lδ3

L1

e1

OO

f // L2

e2

OO

h // L3

e3

OO

Let x ∈ Lδ1; it takes an easy computation to see that

hδfO(x) = hδ lim inf(e2 f)(x) by definition of fO,

= hδ(∨∧

e2 f [F ∩ I] | eF1 (F ) ≤ x ≤ eI1 (I))

by definition of lim inf

=∨∧

hδ e2 f [F ∩ I] | eF1 (F ) ≤ x ≤ eI1 (I)

since hδ preserves∨,∧

,

=∨∧

e3 h f [F ∩ I] | eF1 (F ) ≤ x ≤ eI1 (I)

since hδ e2 = e3 h,

= lim inf(e3 h f)(x) by definition of lim inf,

= (hf)O(x) by definition of (hf)O.

Since x ∈ Lδ1 was arbitrary it follows that hδfO = (hf)O.


If we look at lattice homomorphisms on the right, the situation is morecomplicated. Consider the following picture, where h is a lattice homomorphismand g is an arbitrary map:

L1h // L2

g // L3

When considering the question whether (gh)O = gOhδ, it turns out that it mattersif h is surjective or not. We first record the following observation about the actionof surjective lattice homomorphisms on lattice filters and ideals.

3.2.18. Lemma. Let h : L→M be a surjective lattice homomorphism. Then forall F ∈ F L, I ∈ I L,

F G I ⇒ h[F ∩ I] = F h(F ) ∩ I h(I).

Proof Let F ∈ F L and I ∈ I L and suppose that F G I. Since F ∩ I ⊆ F , wesee that

h[F ∩ I] ⊆ h[F ] ⊆ ↑h[F ] = F h(F ),

and similarly h[F ∩ I] ⊆ I h(I), so that

h[F ∩ I] ⊆ F h(F ) ∩ I h(I).

For the converse, assume that c ∈ F h(F )∩ I h(I) = ↑h[F ]∩ ↓h[I], so there exista ∈ F and b ∈ I such that

h(a) ≤ c ≤ h(b).

Since F G I, by Lemma 2.1.3 we know that F = ↑(F ∩ I) and I = ↓(F ∩ I).Consequently, we may assume without loss of generality that a, b ∈ F ∩ I. Sinceh : L→M is surjective, there must exist some c′ ∈ L1 such that h(c′) = c. Definec′′ := (c′ ∨ a) ∧ b. We will show that h(c′′) = c and that c′′ ∈ F ∩ I, so thatc ∈ h[F ∩ I]. For the first claim, observe that

h(c′′) = h ((c′ ∨ a) ∧ b) by definition,

= (h(c′) ∨ h(a)) ∧ h(b) because h is a homomorphism,

= (c ∨ h(a)) ∧ h(b) because h(c′) = c,

= c because h(a) ≤ c ≤ h(b).

For the second claim, observe that since a ≤ c′ ∨ a, we also get

a ∧ b ≤ (c′ ∨ a) ∧ b = c′′.

Since a, b ∈ F , we also have a ∧ b ∈ F , so that c′′ ∈ F . Since b ∈ I and

c′′ = (c′ ∨ a) ∧ b ≤ b,

we also see that c′′ ∈ I, so that c′′ ∈ F∩I. It follows that F h(F )∩I h(I) ⊆ h[F∩I].


3.2.19. Lemma. Let ei : Li → Lδi be canonical extensions of lattices L1,L2,L3;let h : L1 → L2 be a surjective lattice homomorphism and let g : L2 → L3 be anarbitrary map. Then (gh)O = gOhδ.

Proof Recall that

(gh)O(x) = lim inf(e3 g h)(x)

=∨∧

e3 g h[F ∩ I] | eF1 (F ) ≤ x ≤ eI1 (I)

and

gOhδ(x) =∨∧

e3 g[F ′ ∩ I ′] | eF2 (F ′) ≤ hδ(x) ≤ eI2 (I).

We will show thatg h[F ∩ I] | eF1 (F ) ≤ x ≤ eI1 (I)

=g[F ′ ∩ I ′] | eF2 (F ′) ≤ hδ(x) ≤ eI2 (I ′)

,

(3.15)

which is sufficient to show that (gh)O(x) = gOhδ(x). Take an element of theleft-hand side of (3.15), i.e. take F ∈ F L1, I ∈ I L1 such that eF1 (F ) ≤ x ≤ eI1 (I).We will show that g h[F ∩ I] is an element of the right-hand side of (3.15),because

g h[F ∩ I] = g [F h(F ) ∩ I h(I)] ,

and that eF2 F f(F ) ≤ x ≤ eI2 I h(I). The former follows immediately fromLemma 3.2.18; we see that the latter holds since

eF2 F h(F ) = hδ eF1 (F ) by Lemma 2.2.3(1),

≤ hδ(x) since eF1 (F ) ≤ x ≤ eI1 (I),

≤ hδ eI1 (I) idem,

= eI2 I h(I) by Lemma 2.2.3(2).

Thus we have shown that the left-hand side of (3.15) is contained in the right-handside.

Conversely, consider an element of the right-hand side of (3.15), i.e. takeF ′ ∈ F L2 and I ′ ∈ I L2 such that eF2 (F ′) ≤ hδ(x) ≤ eI2 (I ′). We will show thatg[F ′ ∩ I ′] is an element of the left-hand side of (3.15), because

g[F ′ ∩ I ′] = g h[h−1(F ) ∩ h−1(I)

],

and that eF1 h−1(F ′) ≤ x ≤ eI1 h−1(I ′). The first claim follows from the factthat h is surjective, so that F h and I h are also surjective:

g h[h−1(F ′) ∩ h−1(I ′)

]= g

[F h h−1(F ′) ∩ I h h−1(I ′)

]by Lemma 3.2.18,

= g[F ′ ∩ I ′] by surj. of F h and I h.


For the second claim, note that since eF2 (F ′) ≤ hδ(x) ≤ eI2 (I ′), we have thathδ(x) ∈ ↑ eF2 (F ′) ∩ ↓ eI2 (I ′), so that

x ∈ (hδ)−1(↑ eF2 (F ′) ∩ ↓ eI2 (I ′)

)= (hδ)−1

(↑ eF2 (F ′)

)∩ (hδ)−1 ↓

(eI2 (I ′)

)because (hδ)−1 commutes with ∩,

= ↑ eF1 h−1(F ′) ∩ ↓ eI1 h−1(I ′),

where the last equality follows from claim (2.21) from the proof of Theorem2.2.18(2) since h preserves both binary joins and meets. It follows that theright-hand side of (3.15) is contained in the left-hand side.

Recall the picture we had before, where h is a lattice homomorphism and g isan arbitrary map:

L1h // L2

g // L3

It turns out that if h is not surjective, we need to make several strong assumptionsif we want to prove that (gh)O = gO hδ. For starters, we assume that HSP(L3)is finitely generated.

3.2.20. Lemma. Let ei : Li → Lδi be canonical extensions of lattices L1,L2,L3

and let f : L1 → L2 and g : L2 → L3 be arbitrary maps. Furthermore, assume thatHSP(L3) is finitely generated.

1. If gOfO is (δ, σ↑)-continuous, then gOfO ≤ (gf)O;

2. If gOfO is (δ, σ↓)-continuous, then gOfO ≥ (gf)O;

3. If gOfO is (δ, σ)-continuous, then gOfO = (gf)O.

Proof (1). It is easy to see that gOfO extends gf : L1 → L3, since both squaresbelow commute:

Lδ1fO// Lδ2

gO// Lδ3

L1

e1

OO

f // L2

e2

OO

g // L3

e3

OO

Now since we assumed that gOfO is (δ, σ↑)-continuous, and since ι↑ ⊆ σ↑, it followsthat gOfO ≤ (gf)O, since (gf)O is the largest (δ, ι↑)-continuous extension of gf byCorollary 3.2.15.

(2). If gOfO is (δ, σ↓)-continuous, then

gOfO ≥ (gf)M by Corollary 3.2.15,

≥ (gf)O by Lemma 3.2.8(3).

(3). If gOfO is (δ, σ)-continuous, then a forteriori gOfO is (δ, σ↑)-continuousand (δ, σ↓)-continuous, since σ↑, σ↓ ⊆ σ. The statement now follows by (1) and(2).


We can now straightforwardly apply Lemma 3.2.20 to the case where we havea lattice homomorphism on the right.

3.2.21. Lemma. Let ei : Li → Lδi be canonical extensions of lattices L1,L2,L3; leth : L1 → L2 be a lattice homomorphism and let g : L2 → L3 be an arbitrary map.Furthermore, assume that HSP(L3) is finitely generated. Then gOhδ ≤ (gh)O.

If we additionally assume that gO : Lδ2 → Lδ3 is (σ, σ)-continuous, then gOhδ =(gh)O.

Proof By Theorem 2.2.24, hδ is (δ, δ)-continuous. Since gO is (δ, σ↑)-continuousby Corollary 3.2.8, it follows by general topology that gOhδ is (δ, σ↑)-continuous.Since hδ = hO, it follows by Lemma 3.2.20(1) that gOhδ ≤ (gh)O.

If gO is (σ, σ)-continuous, then by Lemma 2.1.28(3), gO is also (δ, σ)-continuous.By Theorem 2.2.24, hδ is (σ, σ)-continuous, so we see that gOhδ is (δ, σ)-continuous.It follows by Lemma 3.2.20(3) that gOhδ = (gh)O.


In this section we investigated canonical extensions of arbitrary maps betweenlattices, rather than extensions of order-preserving maps (which we studied in§2.2). Canonical extensions of arbitrary maps between distributive lattices havebeen studied extensively by Gehrke and Jonsson [39]. This section is based on ourpaper with M. Gehrke [43]; our contribution lies primarily in two observations:

• The results in [39] hold not only for maps between distributive lattices,but more generally for maps between lattices which lie in finitely generatedvarieties.

• When considering the canonical extensions fO : Lδ →Mδ and fM : Lδ →Mδ

of a map f : L → M, the natural topology one should be using on thecodomain Mδ is the σ↑-topology, respectively the σ↓-topology. This isa departure from [39], where one would have considered the ι↑-topologyand the ι↓-topology on Mδ. Our choice for the σ topologies was inspiredby Y. Venema’s treatment of canonicity for BAOs [89] and the work onMacNeille completions by Theunissen and Venema [86].

Most of the results from §3.2.1 were only known to hold for distributive latticesfrom [39]. What was not known before however, is that one does not need to makeany assumptions about the lattices or the maps involved to prove the results in§3.2.1. In §3.2.2 we introduced topological algebra into the picture of canonicalextensions. Theorem 3.2.12, which says that Lδ is profinite if HSP(L) is finitelygenerated, can be regarded as a corollary of a result of J. Harding [50]. It isinteresting to note that many of the proofs in §3.2.3 are direct adaptations ofthe proofs for distributive lattices from [39], which further supports our claimabove that all results in that paper may be generalized from distributive latticesto lattices lying in a finitely generated variety.

3.3. Canonical extension as a functor II: lattice-based algebras 83

3.2.22. Remark. We now have two sets of circumstances under which fO : Lδ →Mδ is the largest continuous extension of f : L→M (and dually, we have conditionsunder which fM is the smallest continuous extension of f).

• If f is order-preserving, then fO is the largest (δ↑, σ↑)-continuous extensionof f (Theorem 2.2.4).

• If HSP(M) is finitely generated, then fO is the largest (δ, σ↑)-continuousextension of f (Corollary 3.2.15).

It would be interesting to see if there is a unifying explanation for these continuityproperties.

3.3 Canonical extension as a functor II: lattice-

based algebras

In this section, we want to change our perspective on canonical extension from aconstruction on lattices to a construction on lattice-based algebras. Previously,canonical extensions of lattice-based algebras have only been considered undercertain additional assumptions, such as monotonicity of all algebra operations [34],or distributivity of the underlying lattice [39]. In contrast, we will define canonicalextensions for lattice-based algebras without making any further assumptionsabout the algebra operations or the shape of the lattice (other than boundedness).

Once we have defined canonical extensions of lattice-based algebras, it doesnot follow straightforwardly that the canonical extension construction appliedto lattice based-algebras is well-defined on algebra homomorphisms, i.e. whethercanonical extension is a functor. In fact, it is already known from [39] that ingeneral this is not the case, unless we make certain assumptions about either thealgebras or the homomorphisms involved. We will discuss two ways to improvethe behaviour of canonical extensions of homomorphisms. Firstly, if h : A → Bis a homomorphism between lattice-based algebras and every algebra operationis monotone, i.e. order-preserving or order-reversing in each coordinate, thenhδ : Aδ → Bδ is also an algebra homomorphism. This is already known from [34].If we do not know whether all algebra operations on A and B are monotone, but wedo know that h is surjective, then hδ : Aδ → Bδ is also an algebra homomorphism.If the assumption of surjectivity of h is dropped, we can no longer guarantee that hδ

will be a homomorphism. This preservation of surjective algebra homomorphismswas already known for the distributive case from [39]. We will see however thatdistributivity is not needed for this result.

A second question we may ask ourselves is:

For which equations is validity on a lattice-based algebra A preservedwhen moving to its canonical extension Aδ?


This topic is known as canonicity. One could argue that the study of canonicityis actually the raison d’etre of the body of work on canonical extensions thatthis chapter and the previous are contributing to. The approaches to provingcanonicity occupy a spectrum ranging from the imposing of very restrictiveabstract conditions on A (e.g. demanding that HSP(A) is finitely generated)which guarantee preservation of validity of all equations, to sophisticated toolkitsthat allow one to decide whether validity of one given equation is preserved bylooking at the syntactic shape of the given equation. The results on canonicalextensions in this dissertation make a technical contribution to the methods ofproving canonicity at both ends of the above-mentioned spectrum, but remainfoundational. We will not go into any concrete applications of canonicity.

This section is organized as follows. First, we set up the technicalities ofdefining canonical extensions of lattice-based algebras. In particular we needto take some care when extending order-reversing maps, and for every algebraoperation ωA : An → A, we need to choose whether we want its canonical extensionto be ωOA or ωMA . After that we look at preservation of homomorphisms; we concludethis section with a discussion of the relation of this work to the field of canonicity.We discuss the contribution of this section and possible further work on p. 93.

3.3.1 Order types and canonical extension types

Our goal in this subsection is to define canonical extensions for any algebra witha lattice reduct. So let us first make this notion of lattice-based algebra a littlemore precise.

3.3.1. Definition. Let Ω be an algebraic signature and let ar : Ω → N be itsassociated arity function (see §A.6). We say that Ω is a lattice-based similaritytype if ∧,∨, 0, 1 ⊆ Ω; in the remainder of this section we will always assumethat we are dealing with lattice-based signatures. Given a lattice-based similaritytype Ω, a lattice-based Ω-algebra is an Ω-algebra A = 〈A, (ωA)ω∈Ω〉 such that〈A,∧A,∨A, 0A, 1A〉 is a lattice. We denote the lattice reduct of A by Al. We willusually suppress the subscripts on algebra operations, writing ∧ instead of ∧A,etc.

By LatAlgΩ we denote the category of lattice-based Ω-algebras and Ω-algebrahomomorphisms. If it is clear from the context what Ω is, we will simply speak oflattice-based algebras and algebra homomorphisms.

Now, we would like to define canonical extensions of lattice-based algebras insuch a way that we can profit maximally from the results in §2.2. In particular,we want to see order-reversing maps as a variation of order-preserving maps. Thereason this is possible is that a map f : L→ M is order-reversing if and only iff : Lop → M is order-preserving. So if e.g. g : L × L → L is a function that isorder-reversing in its first coordinate and order-preserving in its second coordinate,


then equivalently, g : Lop × L → L is order-preserving. The following definitionwill provide us with notation for dealing with such maps in a uniform way.

3.3.2. Definition. An order type of arity n is an element o ∈ 1, opn. Ifo = (o1, . . . , on) is an order type and L is a lattice, then we define

Lo := Lo1 × · · · × Lon ,

where L1 := L and Lop is the usual order dual of L. If f : L→ M is a function,then we define f o : Lo →Mo as follows:

f o : (x1, . . . , xn)→ (f(x1), . . . , f(xn)) .

It is easy to see that f o is again a lattice homomorphism.Observe that the usual product construction of lattices is a special case of the

above construction: if we take oi = 1 for i = 1, . . . , n, then Lo = Ln and likewisefor lattice homomorphisms.

We can now define monotone lattice-based algebras, i.e. lattice-based alge-bras with operations that are either order-preserving or order-reversing in eachcoordinate.

3.3.3. Definition. Given an algebraic signature Ω, an order signature is afunction ord: Ω → 1, op∗ such that for all ω ∈ Ω, ord(ω) ∈ 1, opar(ω). Anord-monotone lattice based Ω-algebra A is a lattice-based Ω-algebra such that forall ω ∈ Ω, ωA : Aord(ω) → A is order-preserving.

By MLatAlg(Ω,ord) we denote the category of ord-monotone lattice basedΩ-algebras and Ω-algebra homomorphisms. If it is clear from the context what Ωand ord are, we will simply speak of monotone lattice-based algebras.

3.3.4. Example. Heyting algebras are an example of monotone lattice-basedalgebras. Their signature is Ω = →,∧,∨, 0, 1, where ar(→) = 2, and their ordertypes are as follows: ord(→) = (op, 1) and ord(∧) = ord(∨) = (1, 1). Indeed,

∧A : A× A→ A,∨A : A× A→ A,→A : Aop × A→ A,

are all order-preserving.

We know that canonical extensions commute with products and taking orderduals, modulo isomorphism. In practice it is rather cumbersome to explicitly dealwith said isomorphisms all the time however.


3.3.5. Convention. Let L be a lattice and let o be an order type. It follows byLemma 2.1.26 that (Lδ)o is a canonical extension of Lo. In light of this fact, we willidentify (Lδ)o and (Lo)δ. Consequently, if h : L→M is a lattice homomorphism,then we also identify (hδ)o : (Lδ)o → (Mδ)o and (ho)δ : (Lo)δ → (Mo)δ.

We can now finally define canonical extensions of lattice-based algebras.

3.3.6. Definition. Let Ω be a lattice-based similarity type. A canonical exten-sion type for Ω is a function β : Ω→ O,M. Given a lattice-based Ω-algebra A =〈A, (ωA)ω∈Ω〉, the β-canonical extension of A is the algebra Aδ := 〈Aδ, (ωAδ)ω∈Ω〉.For each ω ∈ Ω, the map ωAδ : (Aδ)n → Aδ, where n = ar(ω), is

(ωA)β(ω) :(Aδ)n → Aδ

If it is clear from the context what β is, we will simply speak of the canonicalextension of A.

If A is a monotone lattice-based algebra with respect to some order signatureord, then we define ωAδ as

(ωA)β(ω) :(Aδ)ord(ω) → Aδ.

Observe that we are really using Convention 3.3.5 above: strictly speaking, (ωA)β(ω)

is a map (Ao)δ → Aδ.

We can now finally speak of canonical extensions of algebras. We should keepin mind however that a priori, a lattice-based algebra has many different canonicalextensions: for every algebra operation we may choose either the upper or thelower canonical extension. Sometimes, however, these choices do not arise. Recallthat a map f : L → M between lattices L and M is called smooth if fO = fM.Analogously, we say a lattice-based Ω-algebra A is smooth if for every ω ∈ Ω,(ωA)O = (ωA)M. Observe that in case A is smooth, A has a unique canonicalextension, since it does not matter whether β(ω) = O or β(ω) =M for any ω ∈ Ω.

3.3.7. Lemma. Fix a lattice-based signature Ω and a canonical extension type β.Let A be a lattice-based Ω-algebra such that HSP(Al) is finitely generated. If Aδ isa Boolean topological algebra then A is smooth.

Proof Suppose that Aδ is a Boolean topological algebra; we need to show that forevery ω ∈ Ω, ωA is smooth. Take ω ∈ Ω and without loss of generality, assume thatβ(ω) = O. Because Aδ is a Boolean topological algebra, we know that (ωA)O mustbe (σ, σ)-continuous, since the topology on Aδ is the σ-topology (Fact 3.1.23(1)).It follows by Corollary 3.2.16 that ωA is smooth; this is where we use the factthat HSP(Al) is finitely generated. Since ω ∈ Ω was arbitrary, it follows that A issmooth.


Finite lattice-based algebras in particular satisfy the conditions of the abovelemma, although it is also easy to see directly that if A is a finite lattice-basedalgebra, then A ' Aδ. Just as we did with lattices in Convention 2.1.12, we willtherefore just define Aδ := A in case A is finite.

3.3.8. Convention. If A is a finite lattice-based algebra, then we define Aδ := A.

We conclude this subsection with some observations about (·)l, the forgetfulfunctor from the category of lattice-based Ω-algebras to the category of lattices.

3.3.9. Fact. Let A be a lattice-based Ω-algebra. Then

1. (Aδ)l = (Al)δ;

2. If HSP(A) is a finitely generated variety of Ω-algebras, then HSP(Al) is afinitely generated variety of lattices;

3. If A is a profinite Ω-algebra, then Al is a profinite lattice.

3.3.2 Preservation of homomorphisms

In this subsection we will prove two main results on preservation of algebrahomomorphisms by canonical extensions. These results are very important inlight of one of the main subjects of this chapter: the relation between canonicalextensions and profinite completions. The latter is characterized externally, interms of algebra homomorphisms. Consequently, homomorphisms form a verynatural element of our discourse. Canonical extensions do not behave perfectly onhomomorphisms; interestingly, they do behave well enough.

3.3.10. Theorem ([34]). Fix a lattice-based similarity type Ω, an order typeord and a canonical extension type β. If A and B are monotone lattice-basedalgebras and if h : A→ B is an algebra homomorphism, then hδ : Aδ → Bδ is alsoan algebra homomorphism.

Proof We know by Theorem 2.2.24(1) that hδ : Aδ → Bδ is a lattice homomor-phism which is both (σ, σ)-continuous and (δ, δ)-continuous. To show that it isan algebra homomorphism, consider an arbitrary ω ∈ Ω. Let ord(ω) = o, sothat ωA : Ao → A is an order-preserving map. Finally, without loss of generality,assume that β(ω) = O, so that ωAδ = (ωA)O. By Convention 3.3.5, it suffices toshow that the following diagram commutes:

(Ao)δ

(ho)δ

(ωA)O// A

hδ

(Bo)δ

(ωB)O// B


It follows from Lemma 3.2.17 that

hδ (ωA)O = (h ωA)O. (3.16)

Secondly, since ho is a fortiori a ∧-homomorphism, it follows by Theorem 2.2.18(2)that (ho)δ is (δ↑, δ↑)-continuous, so by Corollary 2.2.23(3), we see that

(ωB)O (ho)δ = (ωB ho)O. (3.17)

Now because h : A→ B is an Ω-algebra homomorphism, we see that hωA = ωBho,so that

hδ (ωA)O = (h ωA)O by (3.16),

= (ωB ho)O since h ωA = ωB ho,= (ωB)O (ho)δ by (3.17).

Since ω ∈ Ω was arbitrary, we conclude that hδ is an Ω-algebra homomorphism.

Thus we see that canonical extension maps algebra homomorphisms to algebrahomomorphisms, provided we are looking at monotone lattice-based algebras.Since all other things we ask of functors (commuting with function composition,preserving the identity function) already follow from Theorem 2.2.24, we get thefollowing result.

3.3.11. Corollary ([34]). Fix a lattice-based similarity type Ω, an order typeord and a canonical extension type β. Then β-canonical extension forms a functorfrom MLatAlg(Ω,ord) to MLatAlg(Ω,ord).

Now we turn to the more general situation where we do not assume that everyoperation of our lattice-based algebras is monotone. The following theorem was al-ready known for distributive lattice-based algebras [39, Theorem 3.7]; interestingly,however, distributivity is not a necessary condition.

3.3.12. Theorem. Fix a lattice-based similarity type Ω and a canonical extensiontype β. Let A and B be lattice-based algebras. If h : A→ B is a surjective algebrahomomorphism, then so is hδ : Aδ → Bδ.

Proof As in the proof of Theorem 3.3.10, we must show that for arbitrary ω ∈ Ω,we have that hδ ωAδ = ωBδ (hδ)n, where n = ar(ω). Without loss of generality,we again assume that β(ω) = O; now as before, showing that

hδ (ωA)O = (ωB)O (hδ)n


boils down to showing that the following diagram commutes; the difference is thatwe do not have to bother with order types.

(An)δ

(hn)δ

(ωA)O// A

hδ

(Bn)δ

(ωB)O// B

It follows from Lemma 3.2.17 that

hδ (ωA)O = (h ωA)O.

For the other direction, observe that since h : A→ B is surjective, so is hn : An →Bn, so by Lemma 3.2.19 we have that

(ωB)O (hn)δ = (ωB hn)O.

Just like in the proof of Theorem 3.3.10, we see that the diagram above commutesbecause h is an Ω-algebra homomorphism, i.e. because h ωA = ωB hn. Sinceω ∈ Ω was arbitrary we see that hδ : Aδ → Bδ is an Ω-algebra homomorphism.

Unfortunately, it is not possible to improve on the above theorem, for [39,Example 3.8] provides an example of a distributive lattice-based algebra B with asubalgebra A such that Aδ is not a subalgebra of Bδ. However, we will see that forthe main result of §3.4.1, our Theorem 3.3.12 gives us just enough to work with.

3.3.3 Canonicity

To conclude this section we will make a few remarks about the preservation of(equations and) inequations by canonical extensions of lattice-based algebras,i.e. canonicity of inequations. This very rich subject is probably the single mostimportant application of canonical extensions in logic. This has to do with therelation between canonical extensions and Stone duality, a subject we will look atin Chapter 4. The exact details of the applications in logic aside, the question iswhether we can show that if a given inequation s 4 t is valid on a lattice-basedalgebra A (see §A.6), then s 4 t is also valid on Aδ. We will very briefly discussthree approaches to this question.

Finitely generated varieties

The first approach is one of brute force. If we assume that HSP(A) is finitelygenerated, then every inequation valid on A is also valid on Aδ. The reason forthis is that under these assumptions, canonical extensions coincide with profinitecompletions; see §3.4.2. It holds for any algebra A that A, the profinite completion


of A, lies in the variety generated by A (see §3.1.2). Therefore, an equation s ≈ tis valid on A iff it is valid on A; since every inequation s 4 t can be encodedas an equation s ∨ t ≈ t, the same holds for inequations. Profinite completionsare always well-behaved in this sense, and if HSP(A) is finitely generated, thisgood behaviour of A is also exhibited by Aδ since the two are isomorphic. Thisgood behaviour comes at a price, however. There are many interesting varieties oflattice-based algebras which are not finitely generated. To name two of the mostillustrious, we mention the variety of Heyting algebras and the variety of modalalgebras.

Our contribution to this first approach of using finitely generated varieties isthat we show in §3.4.2 that this works for arbitrary lattice-based algebras, ratherthan only for monotone lattice-based algebras [34, Corollary 6.9] or distributivelattice-based algebras [39, Corollary 4.6].

Sahlqvist-style theorems

The second approach to proving canonicity focusses on one inequation s 4 t at atime, relying on syntactic criteria on s and t to prove a result. As an example, wegive a proof for a result that goes back to [38]; the form in which we state it iscloser to [34] though. The particular proof we employ (via dcpo algebras) wasfirst presented in [42].

In the proof of the theorem that follows, we want to exploit properties of dcpoalgebras and dcpo presentations. First of all, we have to explain what a dcpoalgebra is. An Ω-dcpo algebra is simply an ordered Ω-algebra A such that 〈A,≤〉is a dcpo and such that each ωA : Aar(ω) → A is Scott-continuous. In §2.3, wehave considered dcpo presentations, which allowed us to describe dcpos in aneconomical fashion. This technique can be extended so that it is also applicableto dcpo algebras.

3.3.13. Definition. An Ω-dcpo algebra presentation consists of a structure 〈P,v, /, (ωP )ω∈Ω〉 such that 〈P,v, /〉 is a dcpo presentation, 〈P, (ωP )ω∈Ω〉 is an Ω-algebraand each ωP : P ar(ω) → P is cover-stable.

As usual, given a definition of a particular kind of algebra presentation, one needsto show that each such presentation actually presents an object. This is exactlywhat the following fact tells us.

3.3.14. Fact ([60]). Let 〈P,v, /, (ωP )ω∈Ω〉 be a dcpo algebra presentation. Sup-pose that 〈P,v, /〉 presents a dcpo D via η : P → D. Then

1. There exist unique Scott-continuous algebra operations ωD on D such thatη : P → D is an Ω-algebra homomorphism;

2. For all inequations s 4 t, if 〈P, (ωP )ω∈Ω〉 |= s 4 t then also 〈D, (ωD)ω∈Ω〉 |=s 4 t.


Now that we have prepared our magic wand, we can go ahead and present ourtheorem. Observe that we make a distinction between the signature of the algebraA and the signature used in the inequation s 4 t that we want to preserve. Thereason for this is that demanding that each operation in the algebra signatureΩ is an operator would restrict the applicability of the theorem to distributivelattices, since it would require that ∧ : A× A→ A is an operator.

3.3.15. Theorem ([34]). Let A be a lattice-based Ω-algebra and let s 4 t be aninequation. If for each ω occurring in s or t, ωA is an operator and ωAδ = (ωA)O,then A |= s 4 t implies Aδ |= s 4 t.

Proof We present the proof of the theorem as found in [42]. Let s(x1, . . . , xn) andt(x1, . . . , xn) be terms and let Ω′ denote the set of function symbols occurring in sor t. We assume that each ω ∈ Ω′ is an operator. Fix any canonical extension typeβ : Ω→ O,M such that β(ω) = O for all ω ∈ Ω′. Now, suppose that A |= s 4 t;we need to show that Aδ |= s 4 t

Since for each ω ∈ Ω′, we assumed ωA is an operator, we see by Lemma 2.3.8that F ωA is cover-stable for each ω ∈ Ω′. It follows that 〈F A,⊇, (F ωA)ω∈Ω′〉is a dcpo algebra presentation. Moreover, the dcpo algebra it presents is theΩ′-reduct of Aδ, since F ωA = (ωA)O for each ω ∈ Ω′ by Lemma 2.3.8. Now byFact 3.3.14, it follows that if we can show that 〈F A,⊇, (F ωA)ω∈Ω′〉 |= s 4 t, thenwe automatically get that Aδ |= s 4 t. But the former is easy to see: becauseeach operation in s and t is order-preserving, it follows from the fact that F is afunctor from the category of partially ordered sets and order-preserving maps tothe category of co-dcpos that

F sA = sF A, (3.18)

and likewise for t. This can be shown by an easy induction on the complexity ofs. (See §A.6 for a reminder about term functions and universal algebra.)

• Suppose that s = xi. Then sA : An → A is simply the i-th projectionfunction πi : An → A, and by Fact A.5.4, F πi : (F A)n → F A is again thei-th projection function.

• Now suppose that s = ω(t1, . . . , tm), where m = ar(ω) and by inductionhypothesis F(ti)A = (ti)F A. Then

F sA

= F (ωA ((t1)A × · · · × (tm)A)) by definition of sA,

= F ωA F ((t1)A × · · · × (tm)A) since F is a functor,

= F ωA (F(t1)A × · · · × F(tm)A) since F preserves finite products,

= F ωA ((t1)F A × · · · × (tm)F A) by induction hypothesis,

= sF A by definition of sF A.


We see that

sF A = F sA by (3.18),

≤ F tA since sA ≤ tA by assumption,

= tF A by (3.18).

Thus, we see that the validity of s 4 t lifts from A to 〈F A,⊇, (F ωA)ω∈Ω′〉 byfunctorial properties of F , and from there to Aδ by Fact 3.3.14.

Results of this kind form an example of Sahlqvist canonicity [79], that is canonicityof inequations based on their syntactic shape; for examples see [34, 40, 39]. Itshould be noted that what we are discussing here is only half of what Sahlqvisttheory is about: Sahlqvist correspendence (see e.g. [19, Ch. 3]) is as important asthe canonicity we have just discussed.

Ad hoc analysis

The third approach is one of ad hoc analysis. In this case, the idea is to proveonly what is needed to show that validity on A of one given inequation s 4 t ispreserved by canonical extensions. First, we need to think of the term functionsinduced by s(x1, . . . , xn) and t(x1, . . . , xn), i.e. sA : An → A and tA : An → A.Validity of the inequation then becomes equivalent to the statement that sA ≤ tA,and the goal becomes to prove that sAδ ≤ tAδ . For the moment, let us assume thatfor every ω ∈ Ω, we have that ωAδ = (ωA)O. If canonical extensions commutedwith composition of arbitrary functions it would now be a breeze to show thatvalidity of s 4 t is preserved, because then we would see that

sAδ = (sA)O ≤ (tA)O = tAδ .

In practice however, we do not know a priori if the equalities sAδ = (sA)O and(tA)O = tAδ hold, or even the inequalities sAδ ≤ (sA)O and (tA)O ≤ tAδ , which wouldalready be sufficient. The ad hoc analysis approach now consists of using specificknowledge about the algebra operations ωA occuring in sA and tA, in conjunctionwith the results about the interaction of function composition and canonicalextensions from §2.2.3 and §3.2.3, to show that sAδ ≤ (sA)O and (tA)O ≤ tAδ . Thisanalysis would also have to take into account whether ωAδ = (ωA)O or ωAδ = (ωA)M

for every ω in s and t.

One can see the Sahlqvist approach to canonicity as an organized version ofthe ad-hoc analysis we sketched above. It should also be noted that there isa limit to the complexity of the inequations for which we can prove canonicityresults, since it is undecidable in general whether a given inequation is canonical[64, Thm. 9.6.1].

3.4. Profinite completion and canonical extension 93


Most of the results in §3.3 are well-known from the work of Gehrke and Harding[34]; the purpose of this section is simply to set straight the definitions of canonicalextensions of lattice-based algebras, expanding upon canonical extensions oflattices. In our discussion of canonicity using our dcpo approach, we present anew proof of a known canonicity theorem (Theorem 3.3.15). The new proof wepresent was first published by M. Gehrke and the author in [42].

Further work

It may be interesting to see if Theorem 3.3.12, which states that surjectivehomomorphisms are preserved by canonical extensions, can be stretched evenfurther, e.g. to ordered algebras with monotone [32] or non-monotone operations.

3.4 Profinite completion and canonical extension

In this section we will explore the connections between canonical extensions andprofinite completions of lattice-based algebras. These connection are very richand in some cases also very intricate.

The most basic connection between canonical extension and profinite com-pletion is Theorem 3.4.1, which states that the profinite completion A of anylattice-based algebra A can be seen as a complete quotient of Aδ, the canonicalextension of A. As a consequence, without making any assumptions about A,we can prove that Aδ has a universal property with respect to profinite algebras(Corollary 3.4.2). This result is completely general and it shows that there isreally a fundamental connection between the canonical extension and the profinitecompletion of a lattice-based algebra. In general, however, Aδ itself is not profinite,unless we make additional assumptions about A. The most extreme assumptionwe can make is that HSP(A) is finitely generated; in this case, Aδ is the profinitecompletion of A (Theorem 3.4.12). A consequence of this is that Aδ ∈ HSP(A) ifHSP(A) is finitely generated, which is a well-known canonicity result.

We conclude the section with two results which sit between the basic connection(profinite completion as a quotient of canonical extension) and the strongestconnection (canonical extensions coinciding with profinite completions in finitelygenerated varieties). It turns out that if we restrict our attention to monotonelattice-based algebras A, then we can prove a universal property of canonicalextensions with respect to Boolean topological monotone lattice-based algebraswith profinite lattice reducts (Theorem 3.4.14), and a theorem characterizingcertain retracts of canonical extensions (Theorem 3.4.16). The prime exampleof such monotone lattice-based algebras is the class of distributive lattices withoperators, which we will revisit in Chapter 4. We discuss the contribution of thissection and possible further work in §3.4.4.


3.4.1 Universal properties of canonical extension

In this subsection we will discuss an interesting property of the canonical extension,namely that for any lattice-based algebra A and any homomorphism f : A→ B toa profinite lattice-based algebra B, there exists a unique complete homomorphismf ′ : Aδ → B such that f ′ eA = f (Corollary 3.4.2).

Aδ

f ′

A

f//

eA>>~~~~~~~~B

We will arrive at this result by first showing that the profinite completion A isa quotient of Aδ; the canonical extension then inherits the universal property ofµA : A→ A.

Let A be a lattice-based algebra. Recall that A, the profinite completion of A,is the limit of the finite quotients of A. These finite quotients are arranged in adiagram 〈A/θ, fθψ〉θ,ψ∈ΦA , where

ΦA = θ ∈ ConA | A/θ finite,

and fθψ : A/θ → A/ψ is defined if θ ⊆ ψ, as

fθψ : a/θ 7→ a/ψ.

The profinite completion of A, denoted A, is the limiting cone over this diagram (see§3.1.2) and it is characterized by the property that for every cone (fθ : B→ A/θ)ΦA ,

there exists a unique map of cones f : B→ A over 〈A/θ, fθψ〉ΦA . This is how we

defined the natural map µA : A→ A, namely, using the fact that (µθ : A→ A/θ)ΦA ,where µθ : a 7→ a/θ, is a cone over the diagram 〈A/θ, fθψ〉ΦA .

3.4.1. Theorem. Let Ω be a lattice-based similarity type and let A be a lattice-based Ω-algebra. Then there exists an Ω-algebra homomorphism νA : Aδ → A suchthat

Aδ

νA

A

eA??

µA// A

1. νA eA = µA;

2. νA is (σ, σ)-continuous, i.e. a complete homomorphism;

3. νA is surjective;

4. νA = lim inf µA = lim supµA;


Proof Let Ω be the similarity type of A, and fix a canonical extension type β.We claim that

(µδθ : Aδ → A/θ)ΦA is a cone over the diagram 〈A/θ, fθψ〉ΦA . (3.19)

Observe that µδθ : Aδ → A/θ is well-defined: since µθ : A → A/θ is a surjectiveΩ-algebra homomorphism, it follows by Theorem 3.3.12 that µδθ : Aδ → (A/θ)δis also an Ω-algebra homomorphism; since (A/θ)δ = A/θ by Convention 3.3.8,we see that indeed µδθ : Aδ → A/θ is an Ω-algebra homomorphism. To show that(µδθ : Aδ → A/θ)ΦA is a cone over 〈A/θ, fθψ〉ΦA , take θ, ψ ∈ ΦA such that θ ⊆ ψ.Since (µθ : A→ A/θ)θ∈ΦA is a cone, we know that fθψ µθ = µψ; we want to showthat it is also the case that fθψ µδθ = µδψ, that is, we want to show that thefollowing diagram commutes:

Aδ

µδθ

µδψ

""FFFFFFFF

A/θfθψ// A/ψ

But this is easy to see:

fθψ µδθ = f δθψ µδθ since (A/θ)δ = A/θ and (A/ψ)δ = A/ψ,

= (fθψ µθ)δ by Th. 2.2.24, since fθψ and µθ are latt. hom.’s,

= µδψ because fθψ µθ = µψ.

Now that we know that (3.19) holds, it follows from the fact that A is thelimit of 〈A/θ, fθψ〉ΦA that there exists a unique map of cones νA : Aδ → A over〈A/θ, fθψ〉ΦA .

(1). We will show that eA : A→ Aδ is a map of cones over 〈A/θ, fθψ〉ΦA . Thatmeans that we must show that for all θ ∈ ΦA, µδθ eA = µθ, i.e. that the followingdiagram commutes.

AeA //

µθ

Aδ

µδθ

A/θ

It is a basic property of canonical extensions of maps that µδθ eA = eA/θ µθ. SinceA/θ is finite, we know that (A/θ)δ = A/θ. But that means that eA/θ = idA/θ, sothat indeed µδθ eA = µθ. Since θ ∈ ΦA was arbitrary, it follows that eA : A→ Aδis a map of cones. Now since νA : Aδ → A is also a map of cones, we obtain a mapof cones νA eA : A → A over 〈A/θ, fθψ〉ΦA . Since A is the limit of 〈A/θ, fθψ〉ΦA

and both µA : A → A and νA eA : A → A are maps of cones over 〈A/θ, fθψ〉ΦA

from A to A, it follows that νA eA = µA.


(2). Since for each θ ∈ ΦA, µθ : A→ A/θ is a lattice homomorphism, it followsby Theorem 2.2.24 that µδθ : Aδ → A/θ is (σ, σ)-continuous. By Fact 3.1.13, the

σ-topology on A is generated by the base

π−1θ (U) | θ ∈ ΦA, U ⊆ A/θ.

Now if we take a basic open set π−1θ (U), then we see that

(νA)−1(π−1θ (U)

)= (πθ νA)−1 (U) by properties of (·)−1,

= (µδθ)−1(U) because νA is a map of cones,

which is a σ-open subset of Aδ since µδθ : Aδ → A/θ is (σ, σ)-continuous. Weconclude that νA is (σ, σ)-continuous.

(3). Given p ∈ K A, we define Fp := a ∈ A | p ≤ µA(a). We claim that

∀p ∈ K A, p =∧µA[Fp]. (3.20)

It is easy to see that p ≤∧µA[Fp]; towards a contradiction suppose that the

converse is not the case, i.e. suppose that∧µA[Fp] p. Then because A is (bi-)

algebraic, there must exist q ∈ K A such that q ≤∧µA[Fp] and q p. The former

tells us

∀a ∈ Fp, q ≤ µA(a).

The latter tells us that ↑ p \ ↑ q 6= ∅. Now ↑ p and ↑ q are σ-clopen (Lemma 3.1.25),so it follows that ↑ p \ ↑ q is σ-clopen. Now since µA[A] is dense in A (Fact 3.1.21),it follows that there must be some a ∈ A such that µA(a) ∈ ↑ p \ ↑ q. But thenp ≤ µA(a), so that a ∈ Fp, and q µA(a), which is a contradiction. It follows

that∧µA[Fp] ≤ p so that (3.20) holds. Now take x ∈ A, then

x =∨

p ∈ K A | p ≤ x

because A is algebraic,

=∨∧µA[Fp] | p ≤ x by (3.20),

=∨∧νA eA[Fp] | p ≤ x by (1),

= νA

(∨∧eA[Fp] | p ≤ x

)since νA is complete by (2).

Since x ∈ A was arbitrary, it follows that νA is surjective.(4). By Fact 3.3.9(3), (A)l is a profinite lattice. Now by (1) and (2) above, we

can apply Corollary 3.2.10 to see that νA = lim inf µA = lim supµA.

Now that we have a unique complete homomorphism νA : Aδ → A from thecanonical extension to the profinite completion, it is not very difficult to showthat eA : A→ Aδ has a universal property with respect to profinite algebras.


3.4.2. Corollary. Let f : A → B be a homomorphism from a lattice-basedalgebra A to a profinite lattice-based algebra B. Then there exists a uniquecomplete homomorphism f ′ : Aδ → B such that f ′ eA = f . In fact, f ′ =lim inf f = lim sup f .

Aδ

f ′

A

eA>>~~~~~~~~

f// B

Proof Assume that we have a homomorphism f : A→ B to a profinite algebraB. By the universal property of µA : A→ A, the profinite completion of A, thereexists a unique continuous, i.e. complete, homomorphism f : A → B such thatf µA = f . Recall from Theorem 3.4.1 that there exists a complete homomorphismνA : Aδ → A such that νA eA = µA. We now define f ′ := f νA. It followsimmediately that f ′ is a complete homomorphism; moreover

f ′ eA = f νA eA by definition of f ′,

= f µA because νA eA = µA,

= f because f µA = f .

Now since f ′ : Aδ → B is a complete, i.e. (σ, σ)-continuous homomorphism andB is profinite, it follows from Corollary 3.2.10 that f ′ = lim inf f = lim sup f , sothat f ′ is unique.

At this point we would like to remind the reader that even though anyhomomorphism f : A → B from a lattice-based algebra A to a profinite lattice-based algebra B can be extended to a continuous f ′ : Aδ → B, Aδ itself need notbe a profinite algebra. In fact, Aδ need not even be a topological lattice, as wewill see in the example below.

3.4.3. Example. We will present an example of a lattice L such that its canonicalextension Lδ is not meet-continuous. Consider the lattice L = 〈L,∧,∨, 0, 1〉 where

L = 0, 1 ∪ aij | i, j ∈ N,

0 is the bottom, 1 is the top, and

aij ≥ akl ⇐⇒ (i+ j ≤ k + l and i ≥ k).

It is not hard to show that the poset 〈L,≤〉 is a lattice. This lattice, see Figure 3.1,is non-distributive by [23, Theorem 3.6], since

1, a20, a11, a00, a02


0

1

a0,0

a0,1

a0,2

a2,0 a1,0

a1,1

x0

x1

x2

x3

Figure 3.1: The lattice M, with its sublattice L denoted by the solid dots.

is a (non-bounded) sublattice of L which is isomorphic to N5 (see [23]). We nowadd the elements xi, i ∈ N to L, with xi ≤ aij for all i, j ∈ N, and xi ≤ xj for alli ≤ j ∈ N, to obtain a new lattice M. We claim that M, depicted in Figure 3.1, isthe canonical extension of its sublattice L. Rather than proving this in detail, weprovide the reader with the following hints:

• M is complete;

• every element of M is a filter element, i.e. for all b ∈M, there exists a filterF ∈ F L such that b =

∧F ;

• every element of L is an ideal element of M, i.e. only the elements xi fori ∈ N are not ideal elements of M.

Armed with these hints it is not hard to prove that M is the canonical extensionof L. To see that Lδ is not meet-continuous note that

a00 ∧ (∞∨i=0

xi) = a00 ∧ 1 = a00

while∞∨i=0

(a00 ∧ xi) =∞∨i=0

x0 = x0.


Now if Lδ were a topological lattice in its σ-topology, then a fortiori ∧ : Lδ×Lδ →Lδ would have to be (σ↑, σ↑)-continuous, i.e. meet-continuous; however we havejust demonstrated that this is not the case. It follows that Lδ is not a topologicallattice in its σ-topology.

We conclude this subsection with a result stating that νA : Aδ → A also has auniversal property, namely with respect to complete homomorphisms f : Aδ → Bfrom Aδ to a profinite algebra B.

3.4.4. Corollary. Fix a canonical extension type β and consider the canonicalextension e : A → Aδ of a lattice-based algebra A. If f : Aδ → B is a completehomomorphism to a profinite lattice-based algebra B, then there exists a uniquecomplete homomorphism f ′ : A→ B such that f ′ νA = f .

Af ′

Aδ

νA??

f// B

Proof Suppose that f : Aδ → B is a complete homomorphism to a profinitelattice-based algebra B. Then f eA : A → B is a homomorphism from A to aprofinite algebra B, so by the universal property of µA : A → A, there exists acomplete homomorphism f ′ : A→ B such that f ′ µA = f eA.

Af ′

A

µA

77ooooooooooooooooeA// Aδ f

// B

We want to show that f ′ νA = f . By Theorem 3.4.1(1), we see that

f eA = f ′ µA = f ′ νA eA.

Since both f : Aδ → B and f ′ νA : Aδ → B are complete homomorphisms agreeingon eA[A], it follows by Corollary 3.4.2 that f = f ′ νA, which is what we wantedto show.

3.4.2 Finitely generated varieties

In §3.4.1, we saw that every homomorphism f : A → B from a lattice-basedalgebra A to a profinite lattice-based algebra B factors through the canonicalextension eA : A → Aδ, but that we cannot expect Aδ itself to be profinite. Inthis subsection we will study sufficient and sometimes necessary conditions forAδ being profinite, namely the condition that HSP(A), the variety generated


by A, is finitely generated. A different approach to canonical extensions in thissetting, using natural extension techniques and focussing on prevarieties ratherthan varieties, can be found in [26].

Recall that a lattice-based Ω-algebra A is smooth if for every ω ∈ Ω, (ωA)O =(ωA)M, and that in case A is smooth, A has a unique canonical extension, since itdoes not matter whether β(ω) = O or β(ω) =M for any ω ∈ Ω.

We will now prove a series of lemmas which will later help us to prove Theorem3.4.10, which says that if HSP(A) is finitely generated, then Aδ is profinite. Theproof of Theorem 3.4.10 uses the observation, due to Jonsson, that if B is a finitealgebra and HSP(B) is congruence distributive, then HSP(B) = HSPB(B), wherePB stands for taking Boolean products (see §A.6). To exploit this fact, we willneed the following technical lemmas about Boolean products, subalgebras andhomomorphic images of lattice-based algebras.

3.4.5. Lemma. Let A be a lattice-based Ω-algebra and let (px : A→ B)x∈X be aBoolean power decomposition of A. If B is finite then Aδ is profinite.

Proof Suppose that A has a Boolean decomposition (px : A→ B)x∈X where B isa finite algebra. Since B is finite, it follows by Fact 2.1.10 that B ' Bδ. We nowknow by Fact 2.1.27 that on the lattice level, BX is a canonical extension of A,where e : A→ BX is defined as

e : a 7→ (px(a))x∈X .

We will now show that

for all ω ∈ Ω, ωBX = (ωA)O = (ωA)M. (3.21)

This observation has two consequences. Firstly, it shows that BX is the (unique)canonical extension of A and that A is smooth. Secondly, since a product ofprofinite algebras is profinite, it follows that Aδ ' BX is a profinite algebra.

So let us show that (3.21) holds. Pick ω ∈ Ω and let n be the arity of ω. Wewill first show that ωBX is a (σ, σ)-continuous extension of ωA. First, observethat it follows from universal algebra that since each px : A→ B is an Ω-algebrahomomorphism, so is e : A→ BX , i.e. the following diagram commutes and ωBX

is indeed an extension of ωA.

(BX)nωBX // BX

Anen

OO

ωA// A

e

OO

Now since BX is profinite (Fact 3.1.15) and BX is a lattice-based algebra, we knowthat ωBX is (σ, σ)-continuous. We would now like to apply Corollary 3.2.16 to


conclude that (3.21) holds, but in order to do that we need to know that HSP(Al)is finitely generated. Since we assumed that A is a Boolean power of B, we knowthat A ∈ HSP(B), so by Fact A.6.2, HSP(A) is finitely generated. It follows byFact 3.3.9 that HSP(Al) is finitely generated; we may now apply Corollary 3.2.16to conclude that (3.21) holds. It follows that Aδ ' BX so that Aδ is profinite.

The next lemma shows that the property of having a profinite canonicalextension is inherited by subalgebras of lattice-based algebras.

3.4.6. Lemma. Let A, B be lattice-based algebras and assume that HSP(Bl) isfinitely generated. If A is a subalgebra of B and if Bδ is profinite, then Aδ is(isomorphic to) a closed subalgebra of Bδ and, consequently, profinite.

Proof Fix a canonical extension type β. Let us denote the embedding of Ainto B by h : A → B. Below, we will show that hδ : Aδ → Bδ is an Ω-algebrahomomorphism, so that Aδ ' h[Aδ] is isomorphic to a subalgebra of Bδ. It thenfollows from Corollary 2.2.25 and Lemma 3.1.28 that h[Aδ] is closed and hence,profinite by Fact 3.1.16.

So let us show that hδ : Aδ → Bδ is an Ω-algebra homomorphism. Take ω ∈ Ωand let n := ar(ω). Without loss of generality, assume that β(ω) = O. We willshow that the following diagram commutes:

(Bδ)n(ωB)O

// Bδ

(Aδ)n(hδ)n

OO

(ωA)O// Aδ

hδ

OO

Since we assumed that Bδ is a profinite algebra, we know that

(ωB)O : (Bδ)n → Bδ is (σ, σ)-continuous. (3.22)

Now we see that

(ωB)O (hδ)n = (ωB)O (hn)δ by Convention 3.3.5,

= (ωB hn)O by Lemma 3.2.21 and (3.22),

= (h ωA)O because h is an Ω-homomorphism,

= hδ (ωA)O by Lemma 3.2.17.

Since ω ∈ Ω was arbitrary, it follows that hδ : Aδ → Bδ is an Ω-algebra homomor-phism.


3.4.7. Remark. In Lemmas 3.3.7, 3.4.5 and 3.4.6 above, we are assuming thatthe lattice-based algebras involved have a lattice reduct lying in a finitely generatedvariety. The reason for this is that we want to be able to apply results that areconsequences of Corollary 3.2.15. If we wanted, however, we could replace theassumptions concerning finitely generated varieties by restricting our results tomonotone lattice-based algebras. We would then have the results that follow fromTheorem 2.2.4 at our disposal, in particular Corollary 2.2.23.

Now that we have studied profiniteness of canonical extensions in relation withBoolean products and subalgebras, the only thing left to consider is homomorphicimages. Here, we need to do a little more work, starting with a technical lemmaabout continuity properties of complete surjective lattice homomorphisms. Observethat the property that the following lemma ascribes to complete surjective latticehomomorphisms h : L→M is weaker than saying that h is an open map: h onlypreserves forward images of open sets of the shape h−1(U), for U ⊆M.

3.4.8. Lemma. Let h : L → M be a complete surjective lattice homomorphismbetween complete lattices L and M. Let U ⊆M. If h−1(U) is σ↑-open (σ↓-open,σ-open) in L, then U is σ↑-open (σ↓-open, σ-open) in M.

Proof We will only treat the case for the σ↑-topology; the other cases follow byorder duality. Observe that since h : L→M is a complete homomorphism, it hasa left adjoint h[ : M→ L. By Fact A.3.3(1), h[ preserves all joins. Moreover, sinceh is surjective, we know that h h[ = idM (Fact A.3.3(2)). Moreover,

∀x ∈ U, h[(x) ∈ h−1(U). (3.23)

After all, if x ∈ U , then because h h[ = idM, we see that h h[(x) = x ∈ U , sothat h[(x) ∈ h−1(U). Now suppose that U ⊆M such that h−1(U) is σ↑-open. Wewill show that U itself is then also σ↑-open. First, we show that U is an upper set.If x ∈ U and x ≤ y, then since h[ is order-preserving, we see that h[(x) ≤ h[(y).By (3.23), we see that h[(x) ∈ h−1(U). Since we assumed that h−1(U) is an upperset, it follows that h[(y) ∈ h−1(U). Now

y = h h[(y) since h h[ = idM,

∈ h[h−1(U)

]since h[(y) ∈ h−1(U),

= U since h is surjective.

It follows that U is an upper set. Next, suppose that S ⊆M is directed and that∨S ∈ U . Then we see that∨

h[[S] = h[(∨S) because h[ preserves all joins,

∈ h−1(U) by (3.23) since∨S ∈ U .


Now because h−1(U) is σ↑-open, there must exist x ∈ S such that h[(x) ∈ h−1(U).Now

x = h h[(x) since h h[ = idM,

∈ h[h−1(U)

]since h[(x) ∈ h−1(U),

= U since h is surjective.

It follows that U is σ↑-open.

We are now ready to state the result about homomorphic images of algebras whichhave a profinite canonical extension. Note that this result is decidedely weakerthan the corresponding results about Boolean products and subalgebras above,since we only get a Boolean topological algebra rather than a profinite algebra.

3.4.9. Lemma. Let A, B be lattice-based algebras. If h : B → A is a surjectiveΩ-algebra homomorphism and if Bδ is profinite, then Aδ is a Boolean topologicalalgebra.

Proof Fix a canonical extension type β. It follows from Lemma 3.2.11 that (Aδ)lis a Boolean topological lattice, so we already know that the σ-topology on Aδis a Boolean topology. What remains to be shown is that Aδ is a topologicalalgebra, i.e. that for each ω ∈ Ω, (ωA)β(ω) : (Aδ)n → Aδ is (σ, σ)-continuous, wheren = ar(ω). Without loss of generality, suppose that β(ω) = O. Consider thefollowing diagram:

(Aδ)n(ωA)O

// Aδ

(Bδ)n(hδ)n

OO

(ωB)O// Bδ

hδ

OO

We know that this diagram commutes because h : B→ A is a surjective Ω-algebrahomomorphism, so Theorem 3.3.12 applies. Let U ⊆ Aδ be σ-open; we want toshow that ((ωA)O)−1 (U) is σ-open. First, observe that(

(hn)δ)−1 ((ωA)O)

−1(U) =

((ωA)O (hn)δ

)−1(U) by properties of (·)−1,

=(hδ (ωB)O

)−1(U) because hδ is an Ω-hom.

Now (ωB)O is (σ, σ)-continuous because Bδ is profinite (Fact 3.1.23(1)), andhδ is (σ, σ)-continuous because h is a lattice homomorphism (Theorem 2.2.24);

consequently, hδ (ωB)O is (σ, σ)-continuous. It follows that(hδ (ωB)O

)−1(U) is

σ-open. Now since (hn)δ is a complete homomorphism, it follows by Lemma 3.4.8that ((ωA)O)−1 (U) is σ-open. Since U ⊆ Aδ was arbitrary, it follows that (ωA)O

is (σ, σ)-continuous. Since ω ∈ Ω was arbitrary, it follows that Aδ is a Booleantopological algebra.


Now that we have our technical lemmas sorted out, we are ready to prove thefirst main theorem of this subsection.

3.4.10. Theorem. If A is a lattice-based algebra such that HSP(A) is finitelygenerated, then Aδ is profinite and hence, A is smooth.

Proof Suppose that B is a finite lattice-based algebra such that A ∈ HSP(B).Because HSP(B) is congruence distributive, we may conclude by Fact A.6.3 thatHSP(B) = HSPB(B). If A ∈ SPB(B), then by Lemmas 3.4.5 and 3.4.6 it followsthat A is profinite. If A ∈ HSPB(B) however, then Lemma 3.4.9 only tells usthat A is a Boolean topological algebra. But, since A ∈ HSP(B) and HSP(B) iscongruence distributive and finitely generated, we may use Fact 3.1.17 to concludethat A is profinite. It follows by Lemma 3.3.7 that A is smooth.

3.4.11. Remark. Observe that the above proof uses Fact 3.1.17, which is apowerful result from [24]. In a recent paper, Gehrke et al. [35] have showedthat any Boolean topological quotient of a profinite algebra is again profinite,employing an argument based on Stone duality.

The following result is a generalization of the main result of [50]. There, thetheorem was stated for monotone lattice-based algebras. It was communicated tous at the time of writing of this chapter (June/July 2010) that Theorem 3.4.12has been independently discovered by M.J. Gouveia [49]; see §3.4.4 for furtherdiscussion.

3.4.12. Theorem. Fix a lattice-based similarity type Ω and let V be a variety oflattice-based Ω-algebras.

1. If V is finitely generated then for every A ∈ V, A eA−→ Aδ is the profinitecompletion of A;

2. If Ω is finite and for every A ∈ V, A eA−→ Aδ is the profinite completion of A,then V is finitely generated.

Proof (1). Let A ∈ V; it follows from Theorem 3.4.10 that Aδ is profinite. Toshow that eA : A→ Aδ is the profinite completion of A, we only have to show thateA : A→ Aδ has the universal property that characterizes the profinite completion,namely, that if B is a profinite algebra and f : A→ B is a homomorphism, thenthere exists a unique continuous homomorphism f ′ : Aδ → B such that f ′ eA = f .But this follows immediately from Corollary 3.4.2.

(2). The proof for the second statement is identical to that in [50]. The proofuses more background knowledge than we assume for the rest of our discourse; weinclude it for cognoscenti. If for every A ∈ V , A eA−→ Aδ is the profinite completionof A, then the natural map from A to A is injective for every A ∈ V, since


eA : A→ Aδ is always injective. It follows by Fact 3.1.20 that V is residually finite.Now since we assumed that Ω is finite, it follows by [61, Theorem 4.1] that thereexists a finite bound on the size of subdirect irreducibles in V ; consequently, V isfinitely generated.

We conclude this subsection with a canonicity result that is well-known inother, less general forms (e.g. [34, Corollary 6.9]).

3.4.13. Corollary. Let A be a lattice-based algebra. If HSP(A) is finitelygenerated, then Aδ ∈ HSP(A); consequently, every equation which is valid on A isalso valid on Aδ.

Proof If HSP(A) is finitely generated, then by Theorem 3.4.12(1), Aδ ' A. SinceA is a subalgebra of a product of quotients of A, we know that A ∈ HSP(A). Thestatement now follows.

3.4.3 Canonical extension and monotone topological alge-bras

We conclude this section with a universal property of canonical extensions withrespect to Boolean topological lattice-based algebras and a corresponding retractiontheorem. Our motivation for studying this universal property comes from modalalgebras. We will see in §4.1 that Boolean topological modal algebras correspond,via Stone duality, to image-finite Kripke frames. In light of this motivating example,we will make two assumptions about the Boolean topological lattice-based algebrasB we are dealing with in this subsection.

• We only consider monotone lattice-based algebras B;

• we only consider Boolean topological lattice-based algebras B such that Bl,the lattice reduct of B, is a profinite lattice.

We will see in §4.1 that modal algebras, and more generally distributive latticeswith operators, indeed satisfy the conditions above. Moreover, modal algebraswill provide us with a clear example why the conditions above are strictly weakerthan assuming that B is profinite.

3.4.14. Theorem. Fix a lattice-based similarity type Ω and a canonical extensiontype β. Let A be a monotone lattice-based Ω-algebra. If f : A → B is an Ω-homomorphism to a Boolean topological monotone lattice-based Ω-algebra B andif Bl is profinite, then there exists a unique complete Ω-algebra homomorphismf ′ : Aδ → B such that f ′ eA = f .

Aδ

f ′

A

eA>>~~~~~~~~

f// B


Proof Suppose that we have an Ω-homomorphism from f : A→ B where A andB are monotone lattice-based algebras and B is a Boolean topological algebra, andsuppose that Bl is profinite. Under these assumptions Corollary 3.4.2 tells us thatthere exists a unique continuous lattice homomorphism f ′ : (Aδ)l → Bl such thatf ′ eA = f . Moreover, by Corollary 3.2.10 we know that

f ′ = lim inf f = lim sup f. (3.24)

We now have to show that f ′ : Aδ → B is in fact an Ω-algebra homomorphism.Let ω ∈ Ω and let n := ar(ω). Without loss of generality, assume that

β(ω) = O. We now want to show that the following diagram commutes:

(Aδ)n(ωA)O

//

(f ′)n

Aδ

f ′

Bn ωB

// B

In order to show that ωB (f ′)n = f ′ (ωA)O, we need to make a few observations.Firstly, observe that

ωB : Bn → B is both Scott- and co-Scott-continuous, (3.25)

so that ωB preserves both directed joins and co-directed meets. Since B is aBoolean topological algebra, we know that ωB is (σ, σ)-continuous. Since ωB ismonotone, it follows by Lemma 2.1.17 that (3.25) holds.

Our second observation is that

ωB fn = f ′ eA ωA. (3.26)

Since f : A → B is an Ω-homomorphism, we know that ωB fn = f ωA. Nowwe may use the fact that f = f ′ eA to conclude that (3.26) holds. Now letx ∈ (Aδ)n = (An)δ; we see that

ωB (f ′)n(x)

= ωB (lim inf f)n (x) by (3.24),

= ωB lim inf(fn)(x) by Lemma 3.2.4,

= ωB(∨∧

fn[F ∩ I] | eFAn(F ) ≤ x ≤ eIAn(I))

by def. of lim inf,

=∨∧

ωB fn[F ∩ I] | eFAn(F ) ≤ x ≤ eIAn(I)

by (3.25) (†),

=∨∧

f ′ eA ωA[F ∩ I] | eFAn(F ) ≤ x ≤ eIAn(I)

by (3.26),

= f ′(∨∧

eA ωA[F ∩ I] | eFAn(F ) ≤ x ≤ eIAn(I))

since f ′ is complete,

= f ′ lim inf(eA ωA)(x) by def. of lim inf

= f ′ (ωA)O(x) by definition of (ωA)O,


where (†) makes use of the fact that F ∩ I is both directed and co-directed forany filter F and ideal I, and the fact that fn : An → Bn is order-preserving, sothat fn[F ∩ I] is again both directed and co-directed.

Since x ∈ (Aδ)n was arbitrary we see that ωB (f ′)n = f ′ (ωA)O. (In caseβ(ω) =M, we would have used the fact that f ′ = lim sup f in the derivation above.)Since ω ∈ Ω was arbitrary, it follows that f ′ : Aδ → B is an Ω-homomorphism.This concludes our proof.

To conclude this subsection, we will prove a theorem that tells us when amonotone lattice-based algebra A with a profinite lattice reduct, i.e. a structurewhich has a reduct which is already a topological algebra, is in fact a topologicalalgebra in its entire signature. Before we come to this theorem, we first need atechnical lemma.

3.4.15. Lemma. Let L and M be complete lattices such that there exist completehomomorphisms g : Lδ → L and h : Mδ → M such that g eL = idL. Then iff : L→M is an order-preserving map so that either hfO = f g or hfM = f g,then f is (σ, σ)-continuous.

LδfO or fM

//

g

Mδ

h

LidL//

eL??~~~~~~~L

f//M

Proof Throughout the proof, we will suppose that h fO = f g; the other caseis order dual. We will first show that f : L→M preserves directed joins. Recallthat by the universal property of ↓L : L → I L, we can factor eL : L → Lδ aseL = eIL ↓L through I L, the ideal completion of L. We may now draw the bigdiagram in Figure 3.2, about which we make two observations. Firstly, we claim

LδfO

//

g

Mδ

h

I L

eIL==

L

↓L==||||||||

idL// L

f//M

Figure 3.2: A closer look at fO via I L.

that

g eIL is (σ↑, σ↑)-continuous. (3.27)


LδfO

//

g

Mδ

h

F L

eFL==zzzzzzzz

L

↑L==

idL// L

f//M

Figure 3.3: A closer look at fO via F L.

To see why, first observe that since g is a complete homomorphism, it alsopreserves all directed joins, so that g is (σ↑, σ↑)-continuous. Since σ↑ ⊆ δ↑ byLemma 2.1.28(3), it follows that g : Lδ → L is (δ↑, σ↑)-continuous. Now sinceeI : I L→ L is (σ↑, δ↑)-continuous by Theorem 2.1.23, it follows that (3.27) holds.Our second observation is that

h fO eIL is (σ↑, σ↑)-continuous. (3.28)

Similarly to g, as we have seen above, it is the case that h : Mδ →M is (σ↑, σ↑)-continuous. Now since fO eIL is (σ↑, σ↑)-continuous by Corollary 2.2.5, it followsthat (3.28) holds.

We can now show that f : L→M is (σ↑, σ↑)-continuous, i.e. that f preservesdirected joins. Let S ⊆ L be directed, then

f(∨S) = f

(∨geIL ↓L[S]

)since g eIL ↓L = idL,

= fgeIL (∨↓L[S]) by (3.27),

= hfOeIL (∨↓L[S]) since h fO = f g,

=∨hfOeIL ↓L[S] by (3.28),

=∨fgeIL ↓L[S] since h fO = f g,

=∨f [S] since g eIL ↓L = idL.

The proof that f preserves co-directed meets is similar. We need to considerthe diagram in Figure 3.3, about which we make the two following familiarobservations. Firstly, for reasons analogous to those for (3.27), it follows that

g eFL is (σ↓, σ↓)-continuous. (3.29)

Secondly, as before with (3.28) we see that

h fO eFL is (σ↓, σ↓)-continuous. (3.30)

Now if S ⊆ L is co-directed, then we can make the same step-by-step argument asabove to show that f(

∧S) =

∧f [S]. Having established that f is both (σ↑, σ↑)-

continuous and (σ↓, σ↓)-continuous, we see that it follows by general topology(Lemma A.7.3) that f : L→M is (σ, σ)-continuous.


In its most general form, we say an algebra A is a retract of an algebra B ifthere exist homomorphisms f : A→ B and g : B→ A such that gf = idA. So thequestion when a lattice-based algebra is a retract of its canonical extension isthe question under what circumstances there exists a homomorphism g : Aδ → Asuch that g eA = idA. We will answer this question below under two additionalrequirements. Firstly, we will only consider complete homomorphisms g : Aδ → A.Secondly, we will assume that A has a profinite lattice reduct.

3.4.16. Theorem. Fix a lattice-based signature Ω and let A be a monotone lattice-based Ω-algebra such that Al is profinite. Then there exists a unique completelattice homomorphism g : (Aδ)l → Al such that g eA = idA. Moreover, thefollowing are equivalent:

1. g : Aδ → A is an Ω-algebra homomorphism;

2. A is a Boolean topological algebra.

Proof Since we assumed that Al is profinite, the existence of a unique completelattice homomorphism g : (Aδ)l → Al extending idA : A→ A follows from Corollary3.4.2. To show that (2) implies (1), we can apply Theorem 3.4.14 to idA : A→ A.

We will now show that (1) implies (2). Since Al is already a Boolean topologicallattice, we only need to show that for each ω ∈ Ω, ωA : Ao → A is continuous,where o is the order type of ω, and the topology we are considering is σ(A), the bi-Scott topology. Since we assumed that g : Aδ → A is an Ω-algebra homomorphism,we know that the following diagram commutes:

(Aδ)oωAδ //

go

Aδ

g

Ao

(eA)o<<yyyyyyyy

idAo// Ao ωA

// A

Now regardless of whether ωAδ = (ωA)O or ωAδ = (ωA)M, Lemma 3.4.15 tells usthat ωA is (σ, σ)-continuous, since ωA is order-preserving and g is a completehomomorphism. Since ω ∈ Ω was arbitrary, it follows that A is a topologicalalgebra.


At this point in the narrative of this dissertation, §3.4 is a high point that is bestenjoyed standing on a lot of ground work developed in Chapters 2 and 3. To alarge extent, Chapters 2 and 3 were shaped to support the individual results in§3.4. Since this section contributes to an active field, we will now put our resultsin some perspective.


The results in §3.4.1 are an improvement of results from [96]. In [96], werelied on duality theory to establish the fundamental connection between thecanonical extension and the profinite completion (Theorem 3.4.1), so we could onlystate the result for distributive lattices with operators. The connection betweencanonical extensions and profinite completions via duality has been studied moreextensively in the cases of Heyting algebras [15, 14, 16] and distributive latticesand Boolean algebras [16]; also see §4.1.4. It is only now that we have extended(the topological) part of the canonical extension theory (in Chapter 2 and §3.2, andwith Theorem 3.3.12), that we are able to state and prove Theorem 3.4.1 in fullgenerality, without using duality theory. The central reasons for the main pointof Theorem 3.4.1, viz. the fact that the profinite completion A of a lattice-basedalgebra A is always a complete quotient of the canonical extension Aδ, are thefact that canonical extensions preserve surjective algebra homomorphisms and thefact that finite algebras are a fixed point of canonical extensions.

In §3.4.2, we set out to improve on J. Harding’s result [50], which says thatprofinite completions and canonical extensions of monotone lattice-based algebrascoincide in finitely generated varieties, using a proof strategy centered around thefact that HSP = HSPB in finitely generated congruence-distributive varieties. Atthe time of writing of this chapter, we learned that M.J. Gouveia has a paper inpress [49] which contains the same result as our Theorem 3.4.12. Unfortunately wecannot presently compare our approach in §3.4.2 with that of [49], because we havenot seen this paper yet. Our results in §3.4.3, concerning universal properties ofcanonical extensions with respect to Boolean topological lattice-based algebras, aresimilar to unpublished work of Gehrke & Harding, who investigated the questionwhen canonical extensions are a reflector when applied to a lattice-based algebras.

Further work

We believe that there is a lot more work to be done on the subject of canonicalextensions and topological algebra. We will name a few possible more concretequestions.

• Even though we know that the canonical extension of a lattice is not alwaysa topological lattice in its σ-topology (Example 3.4.3), this does not rule outthat there is a meaningful way to see the canonical extension of a lattice L asa topological lattice. One possibility would be to consider ‘mixed’ topologieson Lδ, such as σ↑ ∨ δ↓ or σ↓ ∨ δ↑, much like the Lawson topology σ↑ ∨ ι↓ [45].

• Given the fact that Boolean topological lattices can be characterized order-theoretically (Theorem 3.1.26), it would be interesting to see if we can findfurther universal properties of canonical extensions with respect to Booleantopological algebras.

Chapter 4

Duality, profiniteness and completions

In §3.4, we saw that there are several strong connections between canonicalextensions, profinite completions and topological lattice-based algebras. In thischapter, we will see that some of these connections can also be interpreted throughthe discrete duality for distributive lattices with operators (DLO’s).

Discrete duality is one of the ingredients of the ‘double duality diagram’ formodal logic (Fig. 1.1, p. 3) from Chapter 1. For DLO’s, the diagram looks asfollows.

DLO’sextended // relational Priestley

spacesPriestley duality

oo

forgettopology

semi-topological

DLO’s

discrete //

are includedin

OO

orderedKripke frames

dualityoo

We point out two reasons why the discrete duality results we present areinteresting. Firstly, these results provide algebraic insight into interesting classesof (ordered) Kripke frames. Secondly, these results provide spacial insight intointeresting classes of lattice-based algebras. This latter perspective will be theprimary one in this chapter.

This chapter is organized as follows. In §4.1, we discuss a general duality forprofinite DLO’s, and a way to present profinite completions of DLO’s using duality.We will see that profinite DLO’s correspond to hereditarily finite Kripke frames.In §4.2, we provide a brief survey of how the results from §4.1 specialize to knownresults concerning distributive lattices, Boolean algebras and Heyting algebras.Finally, in §4.3, we will restrict our attention to Boolean algebras with operators,

111

112 Chapter 4. Duality, profiniteness and completions

and we will see that in that case, we can also characterize Boolean topologicalalgebras via duality: Boolean topological Boolean algebras with operators aredual to image-finite Kripke frames.

4.1 Dualities for distributive lattices with oper-

ators

In this section, we will describe discrete duality for profinite distributive lat-tices with operators (DLO’s), and we will discuss the relation between canonicalextensions, profinite completions and duality. Recall that distributive latticescorrespond to Priestley spaces via Priestley duality, and that complete, bi-algebraicdistributive lattices correspond to partially ordered sets via discrete duality. Gold-blatt [47] showed that discrete duality for distributive lattices can be extended to aduality relating DLO+, the category of semi-topological DLO’s, to OKFr, the cat-egory of ordered Kripke frames. Moreover, there is a strong and non-coincidentalconnection between canonical extensions of DLO’s and discrete duality: given aDLO A, one can naturally construct an ordered Kripke frame A• via extendedPriestley duality, which is then the discrete dual of Aδ, the canonical extension ofA.

The two main points of this section are the following. Firstly, we show thatPro- DLOf , the category of profinite DLO’s, forms a subcategory of DLO+,the category of semi-topological DLO’s, and that the discrete duality for semi-topological DLO’s restricts to a duality between profinite DLO’s and hereditarilyfinite ordered Kripke frames.

DLO+ ' OKFrop

Pro- DLOf

⊆

' HωOKFrop

⊆

Secondly, we show that A, the profinite completion of a DLO A, corresponds to agenerated subframe of A•, the dual of Aδ.

A

µA

eA

@@@@@@@

Aδ

νA

A•

A HωA•

v

This section is organized as follows. First, in §4.1.1, we introduce semi-topological DLO’s, and we show how semi-topological DLO’s are a generalization

4.1. Dualities for distributive lattices with operators 113

of Boolean topological DLO’s and profinite DLO’s. Next, in §4.1.2, we take acloser look at the category of ordered Kripke frames; specifically, we show howone can construct direted colimits of ordered Kripke frames. In §4.1.3, we usea categorical argument to show that the category of profinite DLO’s is duallyequivalent to the category of hereditarily finite ordered Kripke frames. Finally,in §4.1.4, we show how the relation between canonical extensions and profinitecompletions can be understood using duality. In §4.1.5 we provide conclusionsand suggestions for further work.

4.1.1 Semi-topological DLO’s

In this subsection we will present several facts and results on topological DLO’swhich we will need further on, and which additionally serve to motivate the phrase‘semi-topological DLO’. Let us start by restricting the notion of DLO we willconsider in this section.

4.1.1. Convention. In full generality, a DLO is a monotone distributive lattice-based algebra (see §3.3.1) A for a signature Ω = 0, 1,∧,∨]Ωj]Ωm such that foreach ω ∈ Ωj , ωA : Aord(ω) → A is a normal operator, and ω ∈ Ωm, ωA : Aord(ω) → Ais a dual normal operator. To simplify our presentation, we fix two naturalnumbers n, m and we will only consider DLO’s with a single normal operator♦ : An → A and dual normal operator : Am → A.

4.1.2. Remark. We choose not to explicitly deal with order-reversing operatorsin this chapter, to simplify the presentation. However, all results we presentgeneralize to operators which are order-preserving in some coordinates and order-reversing in others.

4.1.3. Example. Examples of DLO’s are:

• distributive lattices (with no operators);

• Heyting algebras A = 〈A,∧,∨,→, 0, 1〉, as the Heyting implication→ : Aop×A→ A is a dual normal operator;

• Boolean algebras A = 〈A,∧,∨,¬, 0, 1〉, as Boolean negation ¬ : Aop → A isboth a normal operator and a dual normal operator by De Morgan’s laws;

• modal algebras A = 〈A,∧,∨,¬, 0, 1,〉, since these satisfy (x ∧ y) =x ∧y and 1 = 1.

We may now define semi-topological DLO’s, which are the main objects ofstudy in this section, together with ordered Kripke frames (which we will introducein §4.1.2).


4.1.4. Definition. Let A = 〈A,∧,∨, 0, 1,♦,〉 be a DLO. We say A is a semi-topological DLO if

• A is complete,

• A is bi-algebraic,

• ♦ : An → A is a complete operator (preserves all joins in each coordinate),

• : Am → A is a complete dual operator (preserves all meets in each coordi-nate).

By DLO+ we denote the category of semi-topological DLO’s and complete homo-morphisms.

Semi-topological DLO’s are usually called perfect DLO’s in the literature. Wehope that the facts and results in the remainder of this subsection help to convincethe reader that there are good reasons for the name we have chosen. The firstreason for using the phrase ‘semi-topological’ lies in the fact that if we wereto consider DLO’s without any operators, i.e. ‘naked’ distributive lattices, then‘semi-topological’ distributive lattices are topological lattices.

4.1.5. Fact. Let L be a distributive lattice. Then the following are equivalent:

1. L is semi-topological (i.e. complete and bi-algebraic);

2. L is a profinite lattice.

There are many equivalent characterizations of profinite lattices such as the factabove, see e.g. [27, Th. 2.5].

We see now that semi-topological DLO’s come equiped with a natural, Booleantopology, and that ∨ and ∧ are always continuous with respect to this topology.The reason a semi-topological DLO A may fail to be finite lies in the (lack of)continuity properties of ♦ and . We show below that we can expect somecontinuity from ♦ and , however.

4.1.6. Lemma. Let A be a DLO. Then A is a semi-topological DLO if and only if

1. Al is a profinite lattice;

2. ♦ : An → A is (σ↑, σ↑)-continuous;

3. : Am → A is (σ↓, σ↓)-continuous.


Proof Suppose that A is a semi-topological DLO. Firstly, we see that Al, thelattice reduct of A, is complete and bi-algebraic, so by Fact 4.1.5, Al is a profinitelattice. Secondly, since ♦ : An → A preserves all joins in each coordinate, a fortioriit also preserves directed joins in each coordinate. It follows from Fact A.3.4 that♦ is (σ↑, σ↑)-continuous. The argument for is analogous.

Conversely, suppose that A is a DLO satisfying (1), (2) and (3). It followsby Fact 4.1.5 that A is complete and bi-algebraic. Moreover, by Fact A.3.4,♦ : An → A preserves directed joins in each coordinate. But then ♦ preservesboth finite joins (because it is an operator) and directed joins in each coordinate,it follows by Fact A.4.1 that f preserves all joins in each coordinate. An orderdual argument shows that : Am → A preserves all meets in each coordinate;consequently we may conclude that A is a semi-topological DLO.

We will now see that two important classes of topological DLO’s, namelyBoolean topological DLO’s and profinite DLO’s, are semi-topological. This isinteresting for two reasons. Firstly, it tells us that we can apply the duality forsemi-topological DLO’s to Boolean topological DLO’s and profinite DLO’s, sothat we may better understand them. Secondly, the fact that Boolean topologicalDLO’s are semi-topological is a further motivation for the use of the phrase‘semi-topological’.

4.1.7. Definition. By BoolDLO we denote the category of Boolean topologicalDLO’s and continuous homomorphisms. By Pro- DLOf we denote the categoryof profinite DLO’s and continuous homomorphism.

We know for general reasons that Pro- DLOf is a subcategory of BoolDLO(Fact 3.1.13). If we restrict our attention to distributive lattices rather than DLO’s,this inclusion can also be reversed, as was first demonstrated by K. Numakura[72].

4.1.8. Fact. Pro- DLf∼= BoolDL.

We will now show how semi-topological DLO’s can be seen as a generalizationof the profinite DLO’s and Boolean topological DLO’s.

4.1.9. Corollary. BoolDLO is a full subcategory of DLO+. Consequently, sois Pro- DLOf .

Proof We will show that every Boolean topological DLO is a semi-topologicalDLO. Since all continuous DLO homomorphisms are complete homomorphismsby Fact 3.1.23(3), it then follows that BoolDLO is a full subcategory of DLO+.Moreover, since by Fact 3.1.13 every profinite DLO is also a Boolean topologicalDLO, it follows that Pro- DLOf is also a full subcategory of DLO+.

Suppose A is a Boolean topological DLO. Then Al, the lattice reduct of A, isprofinite by Fact 4.1.8. Also, the operation ♦ : An → A is continuous with respect


to the (unique!) topology on A, i.e. with respect to the bi-Scott topology (Fact3.1.23(1)). Since ♦ is order-preserving, it is also (σ↑, σ↑)-continuous (by Lemma2.1.17). By the same argument if follows that is (σ↓, σ↓)-continuous, so we mayconclude by Corollary 4.1.6 that A is a semi-topological DLO. This concludes ourproof.

4.1.2 Colimits of ordered Kripke frames

In this subsection we will introduce ordered Kripke frames, which correspond tosemi-topological DLO’s via discrete duality, and we will show how to constructdirected colimits of ordered Kripke frames. Moreover, we will see that directedunions of Kripke frames are an example of directed colimits.

Ordered Kripke frames

We begin defining OKFr, the category of ordered Kripke frames and boundedmorphisms, which is dually equivalent to DLO+, the category of semitopologicalDLO’s and continuous homomorphisms.

4.1.10. Definition. An ordered Kripke frame [47] consists of a tuple F =〈XF,≤F,RF, QF〉 where

1. 〈XF,≤F〉 is a partial order;

2. RF ⊆ X ×Xn is an (n+ 1)-ary relation such that

(a) ∀x ∈ X, RF[x] := y ∈ Xn | x RF y is a lower set;

(b) ∀y ∈ Xn, x ∈ X | x RF y is an upper set;

3. QF ⊆ X ×Xm is an (m+ 1)-ary relation such that

(a) ∀x ∈ X, QF[x] is a upper set;

(b) ∀y ∈ Xn, x ∈ X | x QF y is a lower set.

Observe that the conditions on RF and QF can be summarized as follows:

≥F ;RF ; (≥F)n = RF and ≤F ;QF ; (≤F)m = QF.

A bounded morphism between two ordered Kripke frames F = 〈XF,≤F, RF, QF〉and G = 〈XG,≤G, RG, QG〉 is a function f : XF → XG such that

1. ∀x, y ∈ F, if x ≤F y then f(x) ≤G f(y) (f is order-preserving);

2. ∀x, y1, . . . , yn ∈ F, if x RF (y1, . . . , yn) then f(x) RG (f(y1), . . . , f(yn)) andsimilarly for Q (f has the forth property);


3. ∀x ∈ F, ∀y′1, . . . , y′n ∈ G, if f(x) RG (y′1, . . . , y′n) then ∃y1, . . . , yn ∈ F such

that x RF (y1, . . . , yn) and f(y1) = y′1, . . . , f(yn) = y′n (f has the backproperty).

By OKFr we denote the category of ordered Kripke frames and bounded mor-phisms. We will sometimes omit the subscripts when specifying a frame, simplywriting F = 〈X,≤, R,Q〉.

For details about the following fact, see e.g. [40, §2.3]. Of the two functorsthat witness this duality, we will only use (·)+ : OKFrop → DLO+, which we willdescribe in more detail in Definition 4.1.21.

4.1.11. Fact. The categories DLO+ and OKfr are dually equivalent.

Directed colimits of ordered Kripke frames

We will now describe directed colimits of ordered Kripke frames. First, we recallwhat colimits are. A cocone to a poset-indexed diagram of ordered Kripke frames〈Fi, fij〉I in OKFr is an I-indexed collection of maps (gi : Fi → F)I with a commoncodomain F such that for all i, j ∈ I with i ≤ j, we have gj fij = gi.

F

Fi

gi

OO

fij// Fj

gj__????????

Let (hi : Fi → G)I be another cocone to 〈Fi, fij〉I and let e : F→ G be a boundedmorphism. We say e is a map of cocones if for all i ∈ I, e gi = hi.

G Feoo

Fi

hi

OO

gi

??

We call (gi : Fi → F)I a limiting cocone if for all (hi : Fi → G)I , there exists aunique map of cocones e : F→ G. We then call F the colimit of 〈Fi, fij〉I , writingF ' lim−→I

Fi. In this chapter, we will only consider directed diagrams of orderedKripke frames, meaning that we will always assume that for all i, j ∈ I, thereexists a k ∈ I such that i, j ≤ k. What is nice about directed colimits of orderedKripke frames is that we can construct them in an elegant way.

Suppose that 〈Fi, fij〉i,j∈I is a directed diagram of ordered Kripke framesFi = 〈Xi,≤i, Ri, Qi〉. We define a relation ∼I on the disjoint union

⊎I Xi: for all

x ∈ Xi and y ∈ Xj,

x ∼I y :⇔ ∃k ≥ i, j, fik(x) = fjk(y).


It is easy to see that ∼I is reflexive and symmetric; moreover, since I is directed,one can also easily show that ∼I is transitive and hence, an equivalence relation.What we have just described is how to construct directed colimits in the categoryof sets. We will now define an ordered Kripke frame structure on the set of∼I-equivalence classes, i.e. on

⊎I Xi/∼I = [x] | ∃i ∈ I, x ∈ Xi. We define the

relations as follows:

[x] ≤I [y] if ∃i ∈ I and x′, y′ ∈ Fi such that x ∼I x′, y ∼I y′ and x′ ≤i y′,

and similarly for R and Q. Now we define⊎I

Fi/∼I := 〈⊎I

Xi/∼I ,≤I , RI , QI〉.

4.1.12. Lemma. Let 〈Fi, fij〉i,j∈I be a directed diagram of ordered Kripke frames.Then

⊎I Fi/∼I is the colimit of 〈Fi, fij〉i,j∈I .

Proof Suppose that we have a cocone for 〈Fi, fij〉i,j∈I , i.e. an ordered Kripkeframe G = 〈Y,≤, R′, Q′〉 and bounded morphisms gi : Fi → G such that for alli ≤ j, gjfij = gi. We want to find a unique bounded morphism g :

⊎I Fi/∼I → G

extending all the gi. We know from basic category theory that the function

g :⊎

Xi/∼I → Y

[x] 7→ gi(x) for x ∈ Fi,

is the unique function extending all the gi. What remains to be shown is that g isa bounded morphism. To see why e.g. the order ≤I is preserved by g, supposethat [x] ≤I [y]. Then there must exist some i ∈ I and x′, y′ ∈ Fi such that x ∼I x′,y ∼I y′ and x′ ≤i y′. Now we see that

g([x]) = g([x′]) since x ∼I x′,= gi(x

′) since x′ ∈ Fi,

≤ gi(y′) since gi is order-preserving,

= g([y′]) since y′ ∈ Fi,

= g([y]) since y ∼I y′.

Similar arguments show that the relations RI and QI are preserved. Finally,suppose that g([x])R′(y′1, . . . , y

′n). We have to find [y1], . . . , [yn] ∈

⊎I Fi/∼I such

that[x]RI([y1], . . . , [yn]) and g([y1]) = y′1, . . . , g([yn]) = y′n.

Let i ∈ I such that x ∈ Fi; then since g([x]) = gi(x) we see that gi(x)R′(y′1, . . . , y′n).

Since gi is a bounded morphism, it follows that there exist y1, . . . , yn ∈ Xi suchthat xRi(y1, . . . , yn) and gi(y1) = y′1, etc. It follows that [x]RI([y1], . . . , [yn]), andsince g([y1]) = gi(y1), etc, we see that g is indeed a bounded morphism. It followsthat

⊎I Fi/∼I is the colimit of 〈Fi, fij〉i,j∈I .


Colimits via generated subframes

We will now see that diagrams of ordered Kripke frames can occur as collectionsof generated subframes of a given frame F, and that directed colimits of suchdiagrams can be computed by simply taking a union. First, we will define generatedsubframes of ordered Kripke frames.

4.1.13. Definition. Given ordered Kripke frames F′ = 〈X ′,≤′, R′, Q′〉 and F =〈X,≤, R,Q〉, we say F′ is a substructure of F if

• X ′ ⊆ X;

• ≤′ = ≤ (X ′ ×X ′) and similarly for R′ and Q′

Observe that if F′ is a substructure of F, then the relations on F′ are all inducedby the underlying set X ′. We may therefore refer to F′ as the substructure ofF induced by X ′. We say F′ is a generated subframe of F (notation: F′ v F) ifadditionally, we have that

• ∀x ∈ X ′, ∀y1, . . . , yn ∈ X, if xR(y1, . . . , yn), then y1, . . . , yn ⊆ X ′, andsimilarly for Q (F′ is closed under R- and Q-successors).

We present a well-known fact which will be of use in §4.1.4. Let f : F′ → F

be a bounded morphism between ordered Kripke frames F = 〈X,≤, R,Q〉 andF′ = 〈X ′,≤′, R′, Q′〉. We define f [F′] to be the substructure of F induced by f [X ′].

4.1.14. Lemma. Let f : F′ → F be a bounded morphism between ordered Kripkeframes. Then f [F′] is a generated subframe of F.

Proof We need to show that f [F′] is a substructure of F, and that f [F′] is closedunder R- and Q-successors. The former follows by the definition of f [F′] above;the latter from the fact that f , as a bounded morphism, has the back property.

It is not hard to see that the generated subframe relation v on OKFr is apartial order. Consequently, we can view a collection of generated subframes as adiagram in the category of ordered Kripke frames and bounded morphisms. LetFi | i ∈ I be a collection of generated subframes of F. We may impose an orderon I by defining i ≤ j :⇔ Fi v Fj. If we denote the embedding from Fi to Fj byfij : Fi → Fj, then 〈Fi, fij〉i,j∈I becomes a diagram of ordered Kripke frames: ifFi v Fj v Fk, then Fi v Fk, so that the following diagrams commutes.

Fjfjk // Fk

Fi

fij

OO

fik

??~~~~~~~~


We call 〈Fi, fij〉i,j∈I the diagram associated with Fi | i ∈ I; whenever possible wewill not refer explicitly to the embedding maps fij . One of the pleasant propertiesof directed diagrams of generated subframes is that we can easily compute theircolimit. If Fi | i ∈ I is a set of generated subframes of F, we denote by⋃Fi | i ∈ I the substructure of F induced by

⋃Xi | i ∈ I.

4.1.15. Lemma. Let F = 〈X,≤, R,Q〉 be an ordered Kripke frame and let Fi |i ∈ I be a collection of generated subframes of F. Then

⋃IFi is also a generated

subframe of F.Moreover, if we assume that Fi | i ∈ I is directed, then

⋃IFi is the colimit

of the diagram associated with Fi | i ∈ I.

Proof Since⋃IFi is a substructure of F by definition, in order to show that

⋃IFi v

F it suffices to show that⋃IFi is closed under R-successors and Q-successors. So

suppose that x ∈⋃IFi and that xR(y1, . . . , yn) for some y1, . . . , yn ∈ F. Then

there must be some i ∈ I such that x ∈ Fi. Now since Fi v F by assumption, wesee that y1, . . . , yn ∈ Fi. It follows that

⋃IFi is closed under R-successors; the

argument for Q-successors is identical.To see that

⋃IFi is the colimit of the diagram associated with Fi | i ∈ I,

suppose we have a co-cone of bounded morphisms (gi : Fi → G)i∈I to some frameG. Saying that (gi : Fi → G)i∈I is a cocone means that whenever x ∈ Fi v Fj,then gj(x) = gi(x).

Fjgj // G

Fi

gi

??

v

But this is precisely sufficient for allowing us to uniquely define a map g :⋃IFi → G

which commutes with all of the gi: given x ∈⋃IFi, pick any i ∈ I such that

x ∈ Fi and map x to gi(x). To see that g :⋃IFi → G is order-preserving and

has the forth-property, i.e. that it preserves each of the relations ≤, R and Q,suppose that e.g. x, y ∈

⋃IFi and x ≤ y. Then there must exist i, j ∈ I such that

x ∈ Fi and y ∈ Fj. Since we assumed that Fi | i ∈ I is directed, there mustexist some k ∈ I such that Fi v Fk and Fj v Fk. But now it follows from the factthat gk : Fk → G is a bounded morphism that g(x) = gk(x) ≤ gk(y) = g(y). Theargument to show that g :

⋃IFi → G has the back-property is identical to that

from the proof of Lemma 4.1.12. It follows that⋃IFi is the colimit of Fi | i ∈ I.

4.1.3 Duality for profinite DLO’s

In this subsection, we will show that the category of profinite distributive latticeswith operators and the category of hereditarily finite ordered Kripke frames are


dually equivalent. Moreover, this duality is the restriction of the duality betweensemi-topological DLO’s and ordered Kripke frames.

DLO+ ' OKFrop

Pro- DLOf

⊆

' HωOKFrop

⊆

We start by defining the category in the lower right hand corner of the abovediagram.

4.1.16. Definition. An ordered Kripke frame F is hereditarily finite if for allx ∈ F, there exists a finite F′ v F such that x ∈ F′. By HωOKFr we denote thefull subcategory of OKFr whose objects are all hereditarily finite ordered Kripkeframes.

One can see the property of being hereditarily finite as a local finitenessproperty. We will see below that we can express this property categorically.

4.1.17. Lemma. Let F be an ordered Kripke frame. The following are equivalent:

1. F is hereditarily finite;

2. F ' lim−→IFi for some directed diagram of finite frames.

Proof (1) ⇒ (2). If F is hereditarily finite then for every x ∈ F, there is a finiteF′ v F such that x ∈ F′. It follows that F =

⋃F′ v F | F′ finite. It follows by

the first part of Lemma 4.1.15 that the diagram associated with F′ v F | F′ finiteis directed: if F0 v F and F1 v F are finite generated subframes of F, then F0 ∪F1

is also a generated subframe of F, which obviously is still finite. It now followsfrom the second part of Lemma 4.1.15 that F is a directed colimit of finite frames.

(2) ⇒ (1). Suppose that 〈Fi, fij〉i,j∈I is a directed diagram of finite orderedKripke frames; we will show that

⊎Fi/∼I is hereditarily finite. This is sufficient

since by Lemma 4.1.12, lim−→IFi '

⊎Fi/∼I . Recall that each Fi can be mapped

to⋃

Fi/∼I by sending x ∈ Fi to [x], the ∼I-equivalence class of x. Let us denotethese maps by hi : Fi →

⊎Fi/∼I . Let [x] ∈

⊎Fi/∼I ; we want to show that there

is a finite F′ v⊎

Fi/∼I such that x ∈ F′. Since [x] ∈⊎

Fi/∼I , we know thatx ∈

⊎Fi, so there is some i ∈ I such that x ∈ Fi. It follows that hi(x) = [x], so

that [x] ∈ hi[Fi]. Now we know from Fact 4.1.14 that hi[Fi] v⊎

Fi/∼I , and sinceFi is finite, so is hi[Fi]. Since [x] ∈

⊎Fi/∼I was arbitrary, it follows that

⊎Fi/∼I

is hereditarily finite.

Now that we have characterized the category of hereditarily finite orderedKripke frames categorically, it is easy to prove the duality result of this subsection.


4.1.18. Theorem. Pro- DLOf∼= (HωOKFr)op.

Proof We already know that DLO+ ' OKFrop (Fact A.8.1). Since Pro- DLOf

is a full subcategory of DLO+ and HωOKFr is a full subcategory of OKFr, itsuffices to show that the functor (·)+ : DLO+ → OKFrop maps profinite DLO’sto hereditarily finite ordered Kripke frames and vice versa for (·)+ : OKFrop →DLO+; we need not concern ourselves with morphisms.

Suppose that A is a profinite DLO, i.e. that A ' lim←−I Ai for some co-directeddiagram 〈Ai, fij〉i,j∈I . We will show that A+ is a hereditarily finite ordered Kripkeframe. Since being a limit is a categorical property, it is stable under categoricalequivalenc. Since (·)+ : DLO+ → OKFrop is a contravariant functor however, wesee that A+ must be a colimit for the diagram 〈(Ai)+, (fij)+〉i,j∈I , and since thearrows are reversed, this diagram is directed rather than co-directed. Since eachAi is finite, we see by Fact A.8.1 that each (Ai)+ is finite as well. We have nowestablished that

A+ ' lim−→I(Ai)+,

for a directed diagram of finite frames 〈(Ai)+, (fij)+〉i,j∈I . It follows by Lemma4.1.17 that A+ is hereditarily finite.

Conversely, if F is hereditarily finite, then by Lemma 4.1.17 we may assumethat F ' lim−→I

Fi for some directed diagram of finite frames 〈Fi, fij〉i,j∈I . It now

follows by an argument analogous to the one above that 〈F+i , f

+ij 〉i,j∈I is a co-

directed diagram of finite DLO’s and that F+ ' lim←−I F+i ; consequently, F+ is

profinite. This concludes our proof.

4.1.4 Profinite completion via duality

In this subsection, we will show that A, the profinite completion of a DLO A,corresponds to the largest hereditarily finite generated subframe of A•, the primefilter frame of A. We illustrate this with the following picture:

A

µA

eA

@@@@@@@

Aδ

νA

A•

A HωA•

v

The left side of the picture above follows from the commutative diagram of Theorem3.4.1, which states that the profinite completion µA : A → A of a lattice-basedalgebra A can be factored through eA : A→ Aδ, the canonical extension of A, viaa surjective complete homomorphism νA : Aδ → A. The right-hand side of thepicture above shows two dual structures which are derived from our DLO A. The


prime filter frame of A, denoted A•, is known to be the discrete dual of Aδ. Whatwe will see below is that A, the profinite completion of A, is dual to HωA•, whichis the largest hereditarily finite generated subframe of A•.

Canonical extensions via duality

Recall that given a DLO A = 〈A,∧,∨, 0, 1,A,♦A〉, we can think of four differentcanonical extensions of A, depending on how we choose to extend A and ♦A:by taking (A)O and (♦A)O, (A)M and (♦A)M, (A)O and (♦A)M, or (A)M and(♦A)O. Out of these four choices, the last one has the best continuity properties ifwe go by Theorem 2.2.18 (1).

4.1.19. Convention. If A = 〈A,∧,∨, 0, 1,A,♦A〉 is a DLO, then we defineAδ = 〈Aδ,∧,∨, 0, 1, (A)M, (♦A)O〉, meaning that we take the lower extension ofthe operator(s) of A and the upper extension of the dual operator(s).

We will now discuss how the canonical extension of a DLO A can be constructedvia A∗, the extended Priestley dual of A. First, one takes A∗ and one stripsthis structure of its topology, resulting in a discrete ordered Kripke frame wedenote by A•, the prime filter frame of A. This construction is in fact a functor(·)• : DLO→ OKFrop.

4.1.20. Definition. The prime filter frame of a DLO A = 〈A,∧,∨, 0, 1,♦A,A〉is defined as follows:

A• := 〈XA• ,≤A• , RA• , QA•〉,

where

• 〈XA• ,≤A•〉 is the set of prime filters of A, ordered by inclusion;

• F RA• (G1, . . . , Gn) iff ♦A[G1, . . . , Gn] ⊆ F ;

• F QA• (G1, . . . , Gm) iff −1A (F ) ⊆

⋃mi=1bi(A,Gi), where

bi(U, V ) := U × · · · × U︸︷︷︸i−1

× V × U × · · · × U︸︷︷︸m−i

.

If f : A→ B is a DLO homomorphism, then f• : B• → B• is defined simply as

f• : F 7→ f−1(F ).

The next step in constructing Aδ from A is to take the complex algebra of A•. Thecomplex algebra functor (·)+ : OKfrop → DLO+ is part of the discrete dualitybetween semi-topological DLO’s and ordered Kripke frames (Fact 4.1.11).


4.1.21. Definition. If F = 〈X,≤, R,Q〉 is an ordered Kripke frame, then wedefine

F+ := 〈Up(X),∩,∪, ∅, X, 〈R〉, [Q]〉,

the complex algebra of F, where

• 〈Down(X),∩,∪, ∅, X〉 is the lattice of lower sets of 〈X,≤〉;

• 〈R〉 : (U1, . . . , Un) 7→ x ∈ X | R[x] G U1 × · · · × Un;

• [Q] : (U1, . . . , Um) 7→ x ∈ X | Q[x] ⊆⋃mi=1bi(X,Ui).

If f : F→ G is a bounded morphism, then we define f+ : G+ → F+ as

f+ : U 7→ f−1(U).

Having defined the functors (·)• : DLO → OKFrop and (·)+ : OKFrop →DLO+, we can now describe the canonical extension of a DLO A using duality.

4.1.22. Fact ([38]). Let A be a DLO. Then e : A→ (A•)+, where

e : a 7→ F ∈ A• | a ∈ F,

is the canonical extension of A.

The above fact is no coincidence; canonical extensions were developed by Jonssonand Tarski [58] for Boolean algebras with operators precisely to study propertiesof duality-based representations such as that in Fact 4.1.22. Note that rather thandefining Aδ first on the lattice reduct of A, and then adding extensions of and♦ as we did in §3.3, we can define the canonical extension of a DLO directly. Thisis because we restricted our notion of canonical extension in Convention 4.1.19.

4.1.23. Convention. For convenience, we redefine eA : A→ Aδ, the canonicalextension of a DLO A, as eA : A→ (A•)+ (see Fact 4.1.22 above) for the remainderof this chapter.

4.1.24. Remark. Observe that the functor (·)+ : OKFrop → DLO+ maps anordered Kripke frame to its collection of lower sets. Some authors define Priestleyduality differently, using upper sets rather than lower sets. This is possible only ifone also adapts the definition of (·)+ : DLO+ → OKFrop. Both approaches aremathematically equivalent, as long as the readers are aware which approach isbeing used.


The dual of the profinite completion

In the introduction to this subsection, we mentioned that the profinite completionof A correspondes to ‘the largest hereditarily finite generated subframe’ of A•.Below, we will make this notion precise.

4.1.25. Lemma. HωOKFr is a co-reflective subcategory of OKFr.

Proof The statement of the lemma boils down to the following. For each frame F

we need to find a hereditarily finite frame F′ and a bounded morphism h : F′ → F,such that whenever f : G→ F is a bounded morphism from a hereditarily finiteframe G to F, then there exists a unique f ′ : G→ F′ such that h f ′ = f .

F

G

f??

f ′// F′

h

OO

Concretely, we will define F′ to be the following generated subframe of F:

HωF =⋃G | G vω F.

Observe HωF is indeed a generated subframe of F by Lemma 4.1.15, and bythe same lemma, we see that HωF is a colimit of a directed diagram of finiteframes. We will take h : HωF → F to simply be the embedding map. Now wemust show that HωF has the desired universal property. What we will show isthat if f : G→ F is a bounded morphism from a hereditarily finite frame G to F,then f [G] v HωF. But this is easy to see: if x ∈ G, then since G is hereditarilyfinite, there exists a finite G′ v G such that x ∈ G′. It follows that f(x) ∈ f [G′].Since G′ is finite, so is f [G′]. Now since also f [G′] v F by Fact 4.1.14, we see that

f(x) ∈ f [G′] v HωF.

Since x ∈ G was arbitrary, it follows that f [G] v HωF. We may therefore takef ′ : G→ HωF to be f with its codomain restricted to HωF. It follows from basicset theory that f ′ is the unique map such that h f ′ = f .

By basic category theory, Hω : OKFr→ HωOKFr is now a functor [69, §IV.3];if f : F → G is a bounded morphism then Hωf : HωF → HωG is simply therestriction of f to HωF.

4.1.26. Remark. Observe that our definition of HωF above leaves open thepossibility that HωF is the empty frame. This will happen if every point inF generates an infinite generated subframe. Not all textbooks on modal logicuniversally agree with us that the empty frame is actually a frame, cf. [19,Definition 1.19].


We can now show how to construct the profinite completion A of a DLO Avia duality.

4.1.27. Theorem. Let A be a DLO. Then m : A→ (HωA•)+, where

m : a 7→ F ∈ HωA• | a ∈ F,

is the profinite completion of A.

Proof Since the profinite completion of A can be characterized via a universalproperty (Fact 3.1.19), it suffices to show the following three things:

1. (HωA•)+ is profinite;

2. m : A→ (HωA•)+ is a DLO algebra homomorphism;

3. for every DLO homomorphism f : A→ B, where B is a profinite DLO, thereexists a unique continuous DLO homomorphism f ′ : (HωA•)+ → B such thatf ′ m = f .

(1). This follows immediately from the fact that Hω : OKFr→ HωOKFr is afunctor and Theorem 4.1.18.

(2). Let us denote the embedding HωA• v A• by h : HωA• → A•. Then

h+ : (A•)+ → (HωA•)+

is a complete DLO homomorphism. Now it is an easy calculation to show that

m = h+ eA; (4.1)

consequently, m : A→ (HωA•)+ is a DLO algebra homomorphism.(3). Suppose that f : A→ B and that B is a profinite DLO. Then by Corollary

3.4.2, there exists a unique continuous homomorphism f ′′ : (A•)+ → B such that

f ′′ eA = f. (4.2)

Now since (·)+ and (·)+ form a duality, there exists a bijection

ϕ : Hom((A•)+,B) Hom(B+,A•) : ϕ−1,

which is natural in A• and B. This bijection gives us a bounded morphismϕ(f ′′) : B+ → A•. Since B+ is image-finite by Theorem 4.1.18, it follows fromLemma 4.1.25 that there exists a unique g : B+ → HωA• such that h g = ϕ(f ′′).

HωA•h

B+

g;;

ϕ(f ′′)// A•

4.2. A brief survey of subcategories of DLO 127

Observe that it is a consequence of the naturality of ϕ−1 that the following diagramcommutes:

HomOKFr(B+, HωA•)ϕ−1//

h

HomDLO+((HωA•)+,B)

h+

HomOKFr(B+,A•)

ϕ−1// HomDLO+((A•)+,B)

Now we see that

f ′′ = ϕ−1 ϕ(f ′′) since ϕ is a bijection,

= ϕ−1(h g) since h g = ϕ(f ′′),

= ϕ−1(g) h+ by naturality of ϕ−1.

Now we define f ′ := ϕ−1(g). Then we see that

f ′ m = f ′ h+ eA by (4.1),

= ϕ−1(g) h+ eA by definition of f ′,

= f ′′ eA since ϕ−1(g) h+ = f ′′,

= f by (4.2).

Moreover, it follows from the fact that g : B+ → HωA• is unique that f ′ = ϕ−1(g)is unique.


The duality for profinite DLO’s we presented in §4.1.3 was inspired by the generalcategorical approach of Johnstone [54, Ch. VI], and the results on the Heytingalgebra case due to G. & N. Bezhanishvili [14]. The dual characterization ofprofinite completion we present in §4.1.4 was inspired by results on the Heytingalgebra case due to G. Bezhanishvili et al. [15]. In [15], the dual of the profinitecompletion of a Heyting algebra is described using the Nachbin order compactifi-cation; this is a different approach than ours, which uses the fact that hereditarilyfinite ordered Kripke frames form a co-reflective subcategory of the category ofordered Kripke frames (Lemma 4.1.25).

An interesting question for further work is whether one can characterizeBoolDLO non-trivially using duality. We will provide a partial answer to thisquestion in §4.3; we revisit this question explicitly in §4.3.3.

4.2 A brief survey of subcategories of DLO

In this section, we will briefly sketch how the two main results of §4.1 specializeto three well-known subcategories of DLO: the categories of distributive lattices,


Boolean algebras and Heyting algebras. We will now recall the main results from§4.1. Firstly, we showed that the discrete duality for semi-topological DLO’srestricts to a duality for profinite DLO’s and hereditarily finite ordered Kripkeframes.

DLO+ ' OKFrop

Pro- DLOf

⊆

' HωOKFrop

⊆

Secondly, the profinite completion A of a DLO A dually corresponds to the largesthereditarily finite generated subframe of A•, the prime filter frame of A.

A

µA

eA

???????

Aδ

νA

' (A•)+

A•

A ' (HωA•)+ HωA•v

Below we will see that in some cases, the diagrams above collapse. Our primeexamples of DLO’s for which the diagrams do not collapse are Heyting algebras,which we will briefly discuss in §4.2.3, and Boolean algebras with operators. Thelatter will be discussed in greater detail in §4.3.

4.2.1. Remark. In our discussion of completions in this section we have notincluded the MacNeille completion (§A.5.2). Duality for MacNeille completionsof Heyting algebras was discussed by Harding & Bezhanishvili [51]. The relationbetween canonical extensions and MacNeille completions of monotone lattice-basedalgebras has been studied by Gehrke, Harding & Venema [36]. Bezhanishvili &Vosmaer [16] discuss the question when canonical extensions, MacNeille comple-tions and profinite completions of distributivel lattices, Heyting algebras andBoolean algebras are isomorphic, using duality. Finally, we would like to pointout the brief discussion of the circumstances under which profinite completionsand MacNeille completions of modal algebras coincide in [97].

4.2.1 Distributive lattices

In the case of distributive lattices, the discrete duality between DLO+ and OKFrboils down to the duality between DL+, the category of complete, bi-algebraicdistributive lattices and complete lattice homomorphisms, and Pos, the categoryof partially ordered sets and order-preserving maps.

4.2.2. Fact. DL+ ' Posop.


If we look at BoolDL ⊆ DL+, the category of Boolean topological distributivelattices and continuous homomorphisms, and Pro- DLf , the category of profinitedistributive lattices and continuous homomorphisms, then it follows from Facts4.1.5 and 4.1.8 that both of these categories are isomorphic to DL+.

4.2.3. Fact. Pro- DLf∼= BoolDL ∼= DL+.

We combine these two facts in the following picture:

DL+ ' Posop

BoolDL

∼=Pro- DLf

∼=

These dualities and isomorphisms have been discussed and rediscovered manytimes; we refer the reader to Johnstone [54, §VI-3] and Davey et al. [27] for moredetails and historical discussion.

Just like the categories DL+ and Pro- DLf collapse, we see that profinitecompletion and the canonical extension of a distributive lattice L are isomorphic.

L

µL

eL

???????

Lδ ' (L•)+ L•

L '

'

(HωL•)+

=

HωL•

=

The fact that νL : Lδ → L is an isomorphism and that νL eL = µL is a corollaryof Theorem 3.4.1, and was first published by G. Bezhanishvili et al. [15] andJ. Harding [50]. The relation between Lδ and L via duality has been studied ingreater detail in [15] and by G. Bezhanishvili and the author in [16]; also see [27].

4.2.2 Boolean algebras

For Boolean algebras, the discrete duality between DLO+ and OKFr boils downto the well-known duality between CABA, the category of complete, atomicBoolean algebras and complete homomorphisms, and Set, the category of setsand functions.

4.2.4. Fact. CABA ' Setop.


The situation with topological Boolean algebras is similar to that for distribu-tive lattices, but it has one important additional ingredient, which is a result dueto D. Papert Strauss [74].

4.2.5. Fact. If A is a compact Hausdorff Boolean algebra, then A is profinite.

This surprising fact contributes to the following collapse of categories:

4.2.6. Fact. Pro- BAf∼= BoolBA ∼= KHausBA ∼= CABA.

We combine the above facts in the following picture:

CABA ' Setop

KHausBA

∼=BoolBA

∼=

Pro- BAf

∼=

For further discussion of these dualities we refer the reader to [54, §VII-1.16].The relation between the canonical extension and the profininte completion

of a given Boolean algebra A = 〈A,∧,∨,¬, 0, 1〉 is the same as in the case ofdistributive lattices: the map νA : Aδ → A from Theorem 3.4.1 is an isomorphism,so that Aδ ' A, and Aδ ' (A•)+, i.e. the canonical extension of A is isomorphicto the complex algebra of the set of ultrafilters of A. Indeed, the embedding of Ainto (A•)+ is what canonical extensions were defined to describe by Jonsson andTarski [58].

4.2.3 Heyting algebras

For Heyting algebras, the discrete duality between DLO+ and OKFr boils downto a duality between HA+, the category of complete bi-algebraic Heyting algebrasand complete homomorphisms, and IntKFr, the category of intuitionistic Kripkeframes, which consists of partially ordered sets and bounded morphisms. Abounded morphism between intuitionistic Kripke frames f : F→ G, where F =〈XF,≤F〉 and G = 〈XG,≤G〉, is a function f : XF → XG such that

↑G f(x) = f [↑F x],

for all x ∈ XF. One can easily show that every bounded morphism betweenintuitionistic Kripke frames is order-preserving; consequently IntKFr is a (non-full!) subcategory of Pos. The duality between HA+ and IntKFr is obtained bysimply restricting the duality between DL+ and Pos.


4.2.7. Fact. HA+ ' IntKFrop.

It is hard to pin down the origin of Fact 4.2.7. One early source is de Jongh& Troelstra [30]; for more categorical detail see G. Bezhanishvili [13]. As far astopological Heyting algebras are concerned, the following is known.

4.2.8. Fact. Let A be a Heyting algebra. The following are equivalent:

1. A is a Boolean topological algebra;

2. A is profinite;

3. A is complete, bi-algebraic and residually finite.

Proof The equivalence of (1) and (2) was established by Johnstone [54, Prop. VI-2.10]. The equivalence of (2) and (3) was established by G. & N. Bezhanishvili[14].

Hereditarily finite intuitionistic Kripke frames are those frames F such that forall x ∈ F, ↑x is finite. We denote the category of hereditarily finite intuitionisticKripke frames by HωIntKFr. The following result, which can be seen as aspecialization of our Theorem 4.1.18, is due to G. & N. Bezhanishvili [14].

4.2.9. Fact. Pro- HAf ' HωIntKFrop.

We summarize the dualities for Heyting algebras we have discussed in the diagrambelow.

HA+ ' IntKFrop

BoolHA

⊆

Pro- HAf '

∼ =

HωIntKFrop

⊆

4.2.10. Remark. We have seen that for distributive lattices and Boolean algebras,there is no difference between semi-topological algebras and topological algebras.We now have our first example of a category of DLO’s for which not everysemi-topological DLO is a Boolean topological DLO; we will demonstrate thisusing duality. Observe that the inclusion HωIntKFrop ⊆ IntKFrop is strict:consider the poset F = 〈N,≤〉 consisting of the natural numbers with their usualordering. The poset F is an intuitionistic Kripke frame, however, for all x ∈ F,↑x is infinite, so F is a fortiori not hereditarily finite. Now since the inclusionHωIntKFrop ⊆ IntKFrop is strict, the inclusion BoolHA ⊆ HA+ is necessarilyalso strict.


Unlike the situation with distributive lattices and Boolean algebras, profinitecompletions of Heyting algebras do not always coincide with canonical extensions[15, Th. 5.2]; consequently, given a Heyting algebra A, we are faced with thegeneral picture:

A

µA

eA

???????

Aδ

νA

' (A•)+

A•

A ' (HωA•)+ HωA•

v

Here, A• is the prime filter frame of A, and HωA• is the generated subframe ofA• induced by the set of points x ∈ A• such that ↑x is finite. These facts, whichcan again be seen as a specialization of results from §4.1, were first established byG. Bezhanishvili et al. [15].

4.3 Duality for topological Boolean algebras

with operators

In the previous section, we considered topological algebras in three categoriesof distributive lattices with operators: distributive lattices, Boolean algebrasand Heyting algebras. In all three of these categories, every Boolean topologicalalgebra is profinite. In this final section of this chapter, we will discuss the categoryof Boolean algebras with operators (BAO’s). This category has the interestingproperty that not every Boolean topological algebra is profinite. Moreover, we cancharacterize Boolean topological BAO’s via duality: every Boolean topologicalBAO is the dual of an image-finite Kripke frame.

Boolean algebras with operators are the original class of lattice-based algebrasfor which canonical extensions were defined by Jonsson and Tarksi [58]. Theyarise naturally as an algebraic semantics for various modal logics [19] and alsomore generally in algebraic logic, e.g. as relation algebras [59]. We know from§4.1 that profinite BAO’s correspond to hereditarily finite Kripke frames. Wewill see that Boolean topological BAO’s correspond to image-finite Kripke frames,i.e. frames in which each point has finitely many successors. From a coalgebraicviewpoint (see Chapter 5), image-finite Kripke frames are a very natural class offrames: they are coalgebras for the finite powerset functor.

This section is organized as follows. In §4.3.1, we establish the main result ofthis section, namely the duality between Boolean topological BAO’s and image-finite Kripke frames, and we briefly discuss a few examples. Next, in §4.3.2, wetake a closer look at the folk result that a Kripke frame F can be embedded in

4.3. Duality for topological Boolean algebras with operators 133

its ultrafilter extension iff F is image-finite, and we show how this result can beinterpreted using duality.

4.3.1 Duality for Boolean topological BAO’s

In this subsection we will show that given a BAO A, the following three things areequivalent: (1) A is a Boolean topological algebra; (2) A is a compact Hausdorffalgebra; (3) A+, the discrete dual of A, is an image-finite frame. Given the factsand results we have previously stated, the proof of this result is very simple. Webegin with some conventions and definitions. A distinguishing property of BAO’sis that operators and dual operators are interdefinable. If A = 〈A,∧,∨,¬, 0, 1,♦A〉is a BAO with a single normal operator ♦A : Am → A, then

A : (a1, . . . , am) 7→ ¬♦A(¬a1, . . . ,¬am)

is a dual normal operator. For this reason, we will afford ourselves the convenienceof only considering BAO’s A = 〈A,∧,∨,¬, 0, 1,A〉 where A is an m-ary dualnormal operator.1

4.3.1. Definition. The category of semi-topological BAO’s BAO+ consists ofcomplete atomic Boolean algebras with operators A = 〈A,∧,∨,¬, 0, 1,A〉, whereA is an m-ary complete dual normal operator, and complete BAO homomor-phisms. The category of Kripke frames KFr consists of structures F = 〈X,R〉,where R ⊆ X × Xm is an (m + 1)-ary relation. A morphism of Kripke framesf : F→ G is a map satisfying the back and forth conditions of Definition 4.1.10,or equivalently, such that

fm[R[x]] = R[f(x)]

for all x ∈ F.

Now that we have made it more precise what we mean by BAO’s and Kripkeframes, it is time for us to introduce image-finite Kripke frames.

4.3.2. Definition. By ImωKFr we denote the full subcategory of KFr whoseobjects are all image-finite Kripke frames. A frame F = 〈X,R〉 is image-finite (orfinitely branching) if for all x ∈ F, R[x] is finite.

4.3.3. Example. Recall that a frame F is hereditarily finite if for every x ∈ F,there exists a finite generated subframe F′ v F such that x ∈ F′. Not everyimage-finite frame is hereditarily finite: consider the frame F := 〈N, S〉, where Sis the successor relation: x S y iff y = x+ 1. The frame F is image-finite, sincefor every x ∈ F, S[x] = x + 1 is a finite set. However, it is not hard to showthat if F′ v F, then F′ must be infinite.

1We might as well have chosen to take an operator rather than a dual operator as primary.Our choice is a matter of taste.


We now arrive at the key lemma which will allow us to prove the duality resultfor Boolean topological DLO’s and image-finite frames.

4.3.4. Lemma. Let F = 〈X,R〉, with R ⊆ Xm+1, be a Kripke frame. Then F isimage-finite iff [R] : P(W )m → P(W ) is Scott-continuous.

Proof For the sake of simplicity we will only treat the case where m = 2. Recallfrom §A.8 that for all U, V ∈ P(X),

[R](U, V ) := x ∈ X | R[x] ⊆ U ×X ∪X × V .

Suppose that 〈X,R〉 is image-finite; we will show that [R] : P(X)×P(X)→ P(X)is Scott-continuous. By Fact A.3.4, it suffices to show that [R] is Scott-continuousin each coordinate. Now suppose that Ui | i ∈ I ⊆ P(X) is a directed collectionof sets and V ∈ P(X) is an arbitrary set, and take x ∈ [R](

⋃IUi, V ). We will show

that x ∈⋃I [R](Ui, V ); since [R] is order-preserving this is sufficient to show that

[R] preserves directed joins in its first coordinate. Now since x ∈ [R](⋃IUi, V ), it

follows that

R[x] ⊆ (⋃IUi)×X ∪X × V by definition of [R],

=⋃I(Ui ×X) ∪X × V by elementary set theory,

=⋃I(Ui ×X ∪X × V ) idem.

Now since Ui | i ∈ I ⊆ P(X) is directed, so is Ui ×X ∪X × V | i ∈ I. SinceR[x] is finite by assumption, there must be some j ∈ I such that

R[x] ⊆ Uj ×X ∪X × V.

Now we see that x ∈ [R](Uj, V ), so consequently x ∈⋃I [R](Ui, V ). Since x was

arbitrary it follows that

[R](⋃IUi, V ) ⊆

⋃I [R](Ui, V ).

Since Ui | i ∈ I was arbitrary, it follows that [R] preserves directed joins in itsfirst coordinate. An analogous argument shows that this also holds for the secondcoordinate; we conclude that [R] is Scott-continuous.

For the converse, suppose that [R] is Scott-continuous. Take x ∈ X; we wantto show that R[x] is finite. It is easy to verify that [R](X, ∅) = X. Let

S := U ⊆ X | X finite,

then X =⋃S. It follows that x ∈ [R](

⋃S, ∅). Now since [R] is Scott-continuous,

there must be some U1 ∈ S such that x ∈ [R](U1, ∅), so that

R[x] ⊆ U1 ×X ∪X × ∅ = U1 ×X.


An analogous argument shows that there must be a U2 ∈ S such that R[x] ⊆ X×U2.But now we see that

R[x] ⊆ U1 ×X ∩X × U2 = U1 × U2,

and since both U1 and U2 are finite, it follows that R[x] must be finite. Sincex ∈ X was arbitrary it follows that 〈X,R〉 is image-finite.

The lemma above equates being image-finite with a continuity property; all thatis left for us to do is to show that this continuity property precisely characterizesBoolean topological BAO’s.

4.3.5. Theorem. KHausBAO ∼= BoolBAO ' ImωKFrop.

Proof To see why KHausBAO ∼= BoolBAO, first observe that BoolBAO isa full subcategory of KHausBAO because every Boolean topological BAO isnecessarily a compact Hausdorff BAO. Moreover, if A = 〈A,∧,∨,¬, 0, 1,A〉 isa compact Hausdorff BAO, then its Boolean algebra reduct 〈A,∧,∨,¬, 0, 1〉 is acompact Hausdorff Boolean algebra. By Fact 4.2.5, 〈A,∧,∨,¬, 0, 1〉 is a profiniteBoolean algebra; consequently the bi-Scott topology on A is a Boolean topology,so that A is a Booleaen topological BAO. Since A was arbitrary it follows thateach compact Hausdorff BAO is a Boolean topological BAO.

We will now show that BoolBAO ' ImωKFrop. Since BoolBAO is a fullsubcategory of BAO+, and since ImωKFr is a full subcategory of KFr, it sufficesto show that (·)+ maps objects of BoolBAO to objects of ImωKFr, and converselythat (·)+ maps objects of ImωKFr to objects of BoolBAO. First, suppose thatA = 〈A,∧,∨,¬, 0, 1,A〉 is a Boolean topological BAO. Then A : Am → A mustbe (σ, σ)-continuous; since A is order-preserving, it follows by Lemma 2.1.17 thatA is (σ↑, σ↑)-continuous. Now by Lemma 4.3.4, A+ is image-finite.

Next, consider an image-finite Kripke frame F = 〈X,R〉. By duality, F+ is asemi-topological BAO, so the Boolean reduct of F+ is a profinite Boolean algebraand [R] : F+ → F+ is (σ↓, σ↓)-continuous by Lemma 4.1.6. Now since we assumedthat F is image-finite, it follows by Lemma 4.3.4 that [R] is (σ↑, σ↑)-continuous;consequently, by Lemma 2.1.17, [R] : F+ → F+ is (σ, σ)-continuous, so that F+ isindeed a Boolean topological algebra. This concludes our proof.


We can now draw the following diagram to present the dualities for topologicalBAO’s.

BAO+ ' KFrop

KHausBAO

⊆

BoolBAO

∼ =

' ImωKFrop

⊆

Pro- BAOf

⊆' HωKFrop

⊆

4.3.6. Remark. Both inclusions HωKFrop ⊆ ImωKFrop ⊆ KFrop are strict; con-sequently, the inclusions Pro- BAOf ⊆ BoolBAO and KHausBAO ⊆ BAO+

are also strict.To see why the inclusions HωKFrop ⊆ ImωKFrop ⊆ KFrop are strict, consider

the frames F := 〈N,≤〉 and G := 〈N, S〉, where xSy iff y = x+ 1. It is not hardto see that F is not image-finite (cf. Example 4.3.3); it is also not hard to see thatG is image-finite but not hereditarily finite.

4.3.7. Remark. Recall from §3.1.1 that for general reasons, there exists a com-pactification functor β : BAO → KHausBAO. Since KHausBAO forms asubcategory of BAO, β : BAO → KHausBAO is a reflector from BAO toKHausBAO. The behaviour of the compactification functor for BAO’s is similarto that of the profinite completion. For instance, it is a corollary of Theorem 3.4.14that for each BAO A, there exists a unique map h : Aδ → βA such that heA = ηA,where ηA : A→ βA is the natural map from A to βA. The compactification βAcan also be described using duality, analogously to Theorem 4.1.27, using the factthat ImωKFr is a co-reflective subcategory of KFr.

4.3.8. Example. Let n be a natural number and define

altn :=∨

0≤j≤n

(

(∧

0≤i<jpi)→ pj

).

An alternative, equivalent form of this axiom is:∧0≤i≤n

♦pi →∨

0≤i<j≤n

♦(pi ∧ pj).

It is known that if A is a BAO such that A |= altn, then Aδ ∈ HSP(A) [11], alsosee the discussion of ‘logics of bounded alternativity’ in [64]. We will show howthis result can be understood in terms of compact Hausdorff BAO’s.

Assume that A |= altn; then it is known (cf. [11]) that for each x ∈ A•, |R[x]| ≤n. A fortiori, A• is image-finite, so Aδ = (A•)+ is a compact Hausdorff BAO by


Theorem 4.3.5. We will argue that in fact, eA : A→ Aδ is the compactification of A.To establish this fact, it suffices to show that for every homomorphism f : A→ B,where B is a compact Hausdorff modal algebra, there exists a unique continuousf ′ : Aδ → B such that f ′ eA = f . This is indeed the case: by Theorem 4.3.5,B is a Boolean topological algebra, so by Theorem 3.4.14, there exists a uniquecontinuous f ′ : Aδ → B such that f ′ eA = f . Since f : A→ B was arbitrary, weconclude that eA : A→ Aδ is indeed the compactification of A. Now by Corollary3.1.7, Aδ ∈ HSP(A).

4.3.2 Ultrafilter extensions of image-finite Kripke frames

The ultrafilter extension of a Kripke frame is an important construction in themodel theory of modal logic [19]; for instance, it used to state the Goldblatt-Thomason theorem. In this subsection we will investigate a folklore result, namelythat a Kripke frame F can be embedded in its ultrafilter extension if and only if F

is image-finite; we will see that this result is essentially a corollary of Theorem3.4.16.

4.3.9. Definition. Given a frame F = 〈W,R〉, we define

ue F := (F+)•.

There is a natural map iF : F→ ue F, defined by

iF : x 7→ W ⊆ X | x ∈ W,

i.e. sending x ∈ X to the principal ultrafilter over X generated by x.

One might ask if iF : F → ue F is a bounded morphism; we will see in Theorem4.3.11 that this is not always the case. Before we can show why, we need to explainthe connection between ultrafilter extensions of Kripke frames and canonicalextensions of BAO’s. Observe that

(ue F)+ =((F+)•

)+by definition of ue,

=(F+)δ

by Convention 4.1.23,

where the BAO F+ is embedded in (ue F)+ via

eF+ : W 7→ F ∈ ue F | W ∈ F. (4.3)

We will now show that there is a natural relation between the two maps iF : F→ue F and eF+ : F+ → (ue F)+. Recall that Al is the lattice reduct of A, and thatif Al is profinite then by Corollary 3.4.2, there exists a unique complete latticehomomorphism g : (Aδ)l → Al such that g eA = idA.


4.3.10. Lemma. Let F be a Kripke frame. Then

(iF)+ :((ue F)+

)l → (F+)l

is a complete Boolean algebra homomorphism and (iF)+ eF+ = idF+.

Proof Since the duality between semi-topological BAO’s and Kripke frames isan extension of the duality between complete Boolean algebras and sets, it followsthat the (set) function iF : F→ ue F corresponds to a complete Boolean algebrahomomorphism

(iF)+ :((ue F)+

)l → (F+)l.

Moreover, an easy calculation shows that (iF)+eF+ = idF+ . Let us denote elementsof (ue F)+ by W ,V for the duration of this proof. Then since (iF)+ := (iF)−1, wesee that

(iF)+ : W 7→ x ∈ F | iF(x) ∈ W. (4.4)

Now let W ∈ F+, then we see that

(iF)+ eF+(W )

= (iF)+ (F ∈ ue F | W ∈ F) by (4.3),

= x ∈ F | iF(x) ∈ F ∈ ue F | W ∈ F by (4.4),

= x ∈ F | W ∈ iF(x) by elementary set theory,

= x ∈ F | x ∈ W by definition of iF,

= W.

It follows that (iF)+ eF+ = idF+ .

We now arrive at the main result of this subsection.

4.3.11. Theorem. Let F = 〈W,R〉 be a Kripke frame. The natural map iF : F→ue F is a bounded morphism iff F is image-finite.

Proof Recall from our discussion above that (F+)δ = (ue F)+. Now if iF : F→ ue F

is a bounded morphism, then

(iF)+ : (ue F)+ → F+

is a BAO homomorphism, and by Lemma 4.3.10, (iF)+ is a complete lattice algebrahomomorphism such that

(iF)+ eF+ = idF+ .

It now follows from Theorem 3.4.16 that F+ is a Boolean topological algebra, soby Theorem 4.3.5, F is image-finite.


For the converse, assume that F is image-finite. By Theorem 3.4.16, thereexists a unique complete lattice homomorphism

g :((ue F)+

)l → (F+)l

such that eF+ g = idF+ . By Lemma 4.3.10 and uniqueness of g, it must be thecase that g = (iF)+. Now since F+ is a Boolean topological algebra by Theorem4.3.5, g : (ue F)+ → F+ is in fact a modal algeba homomorphism. Since we havealready established that g = (iF)+, it follows by duality that iF is a boundedmorphism.


In §4.3.1, we showed that every compact Hausdorff BAO is a Boolean topologicalBAO, and that these BAO’s correspond to image-finite Kripke frames via discreteduality (Theorem 4.3.5). One of the key insights for this result was provided byLemma 4.3.4, which seems to be folklore. Theorem 4.3.11, which states that aKripke frame F embeds in ue F iff F is image finite, also seems to be a folkloreresult. However, the proof we present, using duality for image-finite frames, andreducing the question to whether F+ is a retract of (F+)δ, is new.

Further work

With §4.3.1 we have provided a partial answer to the question from §4.1.5 whetherBoolDLO can be characterized in a meaningful way via duality: if we restrictour attention from DLO’s to BAO’s, then the answer is yes. The questionwhether BoolDLO can be characterized dually in a non-trivial way is still openhowever. (A trivial characterization would be to say that Boolean topologicalDLO’s correspond to ordered Kripke frames F = 〈X,≤, R,Q〉 such that 〈R〉preserves co-directed intersections of lower sets, and [Q] preserves directed unionsof lower sets.)

Another interesting question, which is related to modal logic, is to see whetherone can prove without using duality that if A is a modal algebra such that A |= ψn,then Aδ is a compact Hausdorff algebra, cf. Example 4.3.8.

Chapter 5

Coalgebraic modal logic in point-freetopology

5.1 Introduction

In this chapter we will show how powerlocales, a construction in point-free topology,can be understood and studied via coalgebraic modal logic.

Hyperspaces and powerlocales

The Vietoris hyperspace construction is a topological construction on compactHausdorff spaces, which was introduced in 1922 by L. Vietoris [95] as a general-ization of the Hausdorff metric. Given a topological space X one defines a newtopology τX on KX, the set of compact subsets of X. This new topology τX hasas its basis all sets of the form

∇U1, . . . , Un := F ∈ KX | F ⊆⋃ni=1Ui and ∀i ≤ n, F G Ui,

where U1, . . . , Un ⊆ X is a finite collection of open sets and F G U is notation toindicate that F ∩ U 6= ∅. Alternatively, one can use a subbasis to generate τX ,consisting of subbasic open sets of the shape

(U) := F ∈ KX | F ⊆ U,

and

♦(U) := F ∈ KX | F G U.

To generate the basic open sets ∇U1, . . . , Un from (U) and ♦(U), one can usethe following expression:

∇U1, . . . , Un = (⋃ni=1Ui) ∩

⋂ni=1♦(Ui).

141

142 Chapter 5. Coalgebraic modal logic in point-free topology

In the field of point-free topology, a considerable amount of general topologyhas been recast in a way which makes it more compatible with cosntructivemathematics and topos theory. (Standard references are Johnstone [54] andVickers [91]). The main idea is to study the lattices of open sets of topologicalspaces, rather than their associated sets of points. In other words, it is an approachto topology via algebra, where one studies categories of locales rather than oftopological spaces. Locales have algebraic representations in the form of theirassociated frames. A frame is a complete lattice in which infinite joins distributeover finite meets. In accordance with the algebraic approach which is prevalentthroughout this dissertation, we will only work with frames in this chapter. Wewill be using the word ‘powerlocale’ rather than ‘powerframe’ however, in orderto remain consistent with standard terminology.

Johnstone defines a point-free, syntactic version of the Vietoris powerlocale,using (a) and ♦(a). However he soon also introduces expressions of the shape

(∨A) ∧

∧b∈B♦(b),

where A and B are finite sets, which should remind the reader of the expressionfor ∇U1, . . . , Un above. Nevertheless, the description of the Vietoris powerlocaleusing (a) and ♦(a) is usually taken as primitive, and not without good reason:one can construct the Vietoris powerlocale by first constructing one-sided localescorresponding to the -generators on the one hand and the ♦-generators on theother, and then joining these two one-sided powerlocales to obtain the Vietorispowerlocale [94]. The question remains however, if one can describe the Vietorispowerlocale directly in terms of its basic opens, corresponding to ∇U1, . . . , Un,rather than the subbasic opens (U) and ♦(U). One of the main contributions ofthis chapter is to show that this is indeed possible.

The cover modality and coalgebraic modal logic

The reader may have noticed that the notation using and ♦ above is highlysuggestive of modal logic; this is no coincidence. In Boolean-based modal logic,one can define a ∇-modality which is applied to finite sets of formulas. The∇-modality then has the following semantics. If M = 〈W,R, V 〉 is a Kripke modeland α is a finite set of formulas, then for any state w ∈ W ,

M, w ∇α iff ∀a ∈ α, ∃v ∈ R[w], M, v a and

∀v ∈ R[w], ∃a ∈ α, M, v a.

In classical modal logic, the ∇-modality is equi-expressive with the - and ♦-modalities, using the following translations:

∇α ≡ (∨α) ∧

∧a∈α♦a,

5.1. Introduction 143

and in the other direction, one can use

a ≡ ∇a ∨ ∇∅, and ♦a ≡ ∇a,>.

The ∇ was first introduced as a modality by Barwise & Moss [10] in the study ofcircularity and by Janin & Walukiewicz [53] in the study of the modal µ-calculus.It was in Moss [71] however that the ∇-modality stepped into the spotlight as amodality suitable for coalgebraic modal logic.

A T -coalgebra is simply a function σ : X → TX, where X is the underlyingset of states of the coalgebra, and a T -coalgebra morphism between coalgebrasσ : X → TX and σ : X ′ → TX ′ is simply a function f : X → X ′ such thatTf σ = σ′ f . Aczel [2] introduced T -coalgebras as a means to study transitionsystems. A natural example of such transition systems is provided by the Kripkeframes and Kripke models used in the model theory of propositional modal logic:the category of Kripke frames and bounded morphisms is isomorphic to the categoryof P -coalgebras, where P : Set→ Set is the covariant powerset functor. Universalcoalgebra was later introduced by J. Rutten [78] as a theoretical framework formodelling behaviour of set-based transition systems, parametric in their transitionfunctor T : Set→ Set.

Coalgebraic logics are designed and studied in order to reason formally aboutcoalgebras; one of the main applications of this approach is the design of specifi-cation and verification languages for coalgebras, i.e. for transition systems. Oneinfluential approach to coalgebraic logic, known as coalgebraic modal logic, is totry and generalize properties of propositional modal logic, because the Kripkesemantics of classical modal logic is coalgebra in disguise. The cover modality ∇was introduced in coalgebraic modal logic by L. Moss [71], who made the funda-mental observation that the modal semantics for ∇ can be described using theset-theoretical/categorical technique of relation lifting. The most recent sound andcomplete derivation system for coalgebraic ∇-logic, the Carioca axiomatization,was introduced through the collaborative efforts of Bılkova, Palmigano & Venema[17] and Kupke, Kurze & Venema [66].

Contribution

The results in this chapter are all joint work with Yde Venema and Steve Vickers.We list some of the main contributions of this work:

• We introduce a generalized powerlocale construction, parametric in a tran-sition functor T . The classical Vietoris powerlocale construction is aninstantiation of the T -powerlocale, where we take T = Pω, the covariantfinite power set functor.

• We show that the connection between the Vietoris construction and thecover modality, which was implicit in semantic form already from Vietoris’s


1922 paper [95], can also be made explicit syntactically using coalgebraicmodal logic. Our approach shows how to describe the Vietoris constructionssyntactically using the∇-expressions as primitives, rather than as expressionsderived from - and ♦-primitives, as it was introduced in [54]. This approachruns parallel to that of Kupke, Kurz & Venema who introduced the Booleanalgebra version of the construction we apply to frames [66, 67].

• Additionally, we take first steps towards developing a geometric coalgebraicmodal logic, i.e. a logic using finite conjunctions and infinite disjunctions.This is a step away from previous work on the Carioca axioms [73, 17, 66, 62,67] where one only considered finite disjunctions. Possible future applicationsin coalgebra include the development of modal logics for coalgebras oncompact Hausdorff spaces, rather than on discrete sets.

This chapter is organized in a more self-contained manner than the rest of thisdissertation. In §5.2 we introduce preliminaries on category theory, relation lifting,frame presentations and the classical point-free presentation of the powerlocale,along with a new compactness proof. In §5.3 we introduce the T -powerlocalefunctor VT . We then show that the Pω-powerlocale is isomorphic to the classical Vi-etoris powerlocale and we demonstrate how one can extend natural transformationsbetween transition functors to natural transformations between T -powerlocalefunctors. We conclude the section with a different presentation of T -powerlocales,which reveals that each element of a T -powerlocale has a disjunctive normal form.Finally in §5.4 we show several first preservation results: we show that VT preservesregularity and zero-dimensionality, and the combination of zero-dimensionalityand compactness.

5.2 Preliminaries

5.2.1 Basic mathematics

First we fix some mathematical notation and terminology, which will sometimesdiffer from that used previously in this dissertation.

Let f : X → X ′ be a function. Then the graph of f is the relation

Grf ::= (x, f(x)) ∈ X ×X ′ | x ∈ X

Given a relation R ⊆ X ×X ′, we denote the domain and range of R by dom(R)and rng(R), respectively. Given subsets Y ⊆ X, Y ⊆ X ′, the restriction of R toY and Y ′ is given as

RY×Y ′ ::= R ∩ (Y × Y ′).The composition of two relations R ⊆ X ×X ′ and R′ ⊆ X ′ ×X ′′ is denoted byR ;R′, whereas the composition of two functions f : X → X ′ and f ′ : X ′ → X ′′ isdenoted by f ′ f . Thus, we have Gr(f ′ f) = Grf ; Grf ′.

5.2. Preliminaries 145

We will denote by P (X) and Pω(X) the power set and finite power set of agiven set X. The diagonal on X is the relation ∆X = (x, x) | x ∈ X. Given twosets X, Y we say that X and Y overlap, notation: X G Y , if X ∩ Y is inhabited(that is, non-empty).

A pre-order is a pair (X,R) where R is a reflexive and transitive relation onX. Given such a pre-order we define the operations ↓(X,R), ↑(X,R) : PX → PXby ↓(Y ) := x ∈ X | x R y for some y ∈ Y and ↑(Y ) := x ∈ X | y Rx for some y ∈ Y . If no confusion is likely, we will write ↓X or ↓ rather than↓(X,R).

5.2.2 Category theory

We will assume familiarity with the basic notions from category theory discussedin §A.2, including those of categories, functors, natural transformations, and (co-)monads.

We let Set denote the category with sets as objects and functions as morphisms;endofunctors on this category will simply be called set functors. The mostimportant set functor that we shall use is the covariant power set functor P , whichis in fact (part) of a monad (P, µ, η), with ηX : X → P (X) denoting the singletonmap ηX : x 7→ x, and µX : PPX → PX denoting union, µX(A) :=

⋃A. The

contravariant power set functor will be denoted by P .

We will restrict our attention to set functors satisfying certain properties, ofwhich the first one is crucial. In order to define it, we need to recall the notion of a(weak) pullback. Given two functions f0 : X0 → X, f1 : X1 → X, a weak pullbackis a set P , together with two functions pi : P → Xi such that f0 p0 = f1 p1,and in addition, for every triple (Q, q0, q1) also satisfying f0 q0 = f1 q1, there isan arrow h : Q→ P such that q0 = h p0 and q1 = h p0, in a diagram:

Q

q0

q1

$$

h

@@

@@

Pp1 //

p0

X1

f1

X0 f0// X

For (P, p0, p1) to be a pullback, we require in addition the arrow h to be unique.

A functor T preserves weak pullbacks if it transforms every weak pullback(P, p0, p1) for f0 and f1 into a weak pullback (TP, Tp0, Tp1) for Tf0 and Tf1. Anequivalent characterization is to require T to weakly preserve pullbacks, that is, toturn pullbacks into weak pullbacks. In the next subsection we will see yet another,and motivating, characterization of this property.

The second property that we will impose on our set functors is that of standard-ness. Given two sets X and X ′ such that X ⊆ X ′, let ιX,X′ denote the inclusion


map from X into X ′. A weak pullback-preserving set functor T is standard if itpreserves inclusions, that is: TιX,X′ = ιTX,TX′ for every inclusion map ιX,X′ .

5.2.1. Remark. Unfortunately the definition of standardness is not uniformthroughout the literature. Our definition of standardness is taken from Moss [71],while for instance Adamek & Trnkova [4] have an additional condition involvingso-called distinguished points. Fortunately, the two definitions are equivalent incase the functor preserves weak pullbacks, see Kupke [65, Lemma A.2.12].

The restriction to standard functors is not essential, since every set functoris ‘almost standard’ [4, Theorem III.4.5]: given an arbitrary set functor T , wemay find a standard set functor T ′ such that the restriction of T and T ′ to allnon-empty sets and non-empty functions are naturally isomorphic.

Finally, we shall require that our functors are determined by their behaviouron finite sets. Call a standard set functor T finitary if TX =

⋃TX ′ | X ′ ⊆ω X.

Our focus on finitary functors is not so much a restriction as a convenient wayto express the fact that we are interested in the finitary version of an arbitraryset functor, in the sense that Pω is the finitary version of P . Generally, we maydefine, for a standard functor T , the functor Tω that on objects X is defined byTωX =

⋃TX ′ | X ′ ⊆ X, while on arrows f we simply put Tωf := Tf .

5.2.2. Convention. Throughout this chapter we will assume that T is a finitary,standard endofunctor on Set that preserves weak pullbacks.

Many set functors satisfy the conditions listed in Convention 5.2.2, guaranteeinga wide scope for the results in this chapter.

5.2.3. Example. The identity functor Id , the finitary power set functor Pω, and,for each set Q, the constant functor CQ (given by CQX = Q and CQf = idQ) arestandard, finitary, and preserve weak pullbacks.

For a slightly more involved example, consider the finitary multiset functorMω. This functor takes a set X to the collection MωX of maps µ : X → N offinite support (that is, for which the set Supp(µ) := x ∈ X | µ(x) > 0 is finite),while its action on arrows is defined as follows. Given an arrow f : X → X ′ and amap µ ∈MωX, we define (Mωf)(µ) : X ′ → N by putting

(Mωf)(µ)(x′) :=∑µ(x) | f(x) = x′.

With this definition, the functor is not standard, but we may ‘standardize’ itby representing any map µ : X → N of finite support by its ‘support graph’(x, µx) | µx > 0. As a variant of Mω, consider the finitary probability functorDω, where DωX = δ : X → [0, 1] | Supp(δ) is finite and

∑x∈X δ(x) = 1, while

the action of Dω on arrows is just like that of Mω.


Perhaps more importantly, the class of finitary, standard functors that preserveweak pullbacks is closed under the following operations: composition () , product(×), co-product (+), and exponentiation with respect to some set D ((·)D). As acorollary, inductively define the following class EKPF ω of extended finitary Kripkepolynomial functors :

T ::= Id | Pω | CQ |Mω | Dω | T0 T1 | T0 + T1 | T0 × T1 | TD.

Then each extended Kripke polynomial functor falls in the scope of the work inthis chapter.

As running examples in this chapter we will often take the binary tree functorB = Id × Id , and the finitary power set functor Pω.

An interesting result of standard functors is that they preserve finite inter-sections [4, Theorem III.4.6]: T (X ∩ Y ) = TX ∩ TY . As a consequence, if T isfinitary, for any object ξ ∈ TX we may define

BaseTX(ξ) :=⋂X ′ ∈ Pω(X) | ξ ∈ TX ′,

and show that BaseTX(ξ) is the smallest set X such that ξ ∈ TX [90]. In fact, thebase maps provide a natural transformation Base : T → Pω; for referencing wewill mention this fact explicitly in the next section.

To facilitate the reasoning in this chapter, which will involve objects of variousdifferent types, we use a variable naming convention.

5.2.4. Convention. Let X be a set and let T : Set → Set be a functor. Weuse the following naming convention:

Set ElementsX a, b, . . . , x, y, . . .

TX α, β, . . .PX A,B, . . .

PTX Γ,∆, . . .TPX Φ,Ψ, . . .

5.2.3 Relation lifting

In §5.1, we mentioned that coalgebraic modal logic using the cover modality, asintroduced by Moss, crucially uses relation lifting, both for its syntax and semantics.Relation lifting is a technique which allows one to extend a functor T : Set→ Setdefined on the category of sets (satisfying the conditions of Convention 5.2.2) toa functor T : Rel→ Rel on the category of sets and relations in a natural way.In this subsection we will introduce some of the basic facts and definitions aboutrelation lifting.


Let T be a set functor. Given two sets X and X ′, and a binary relation Rbetween X and X ′, we define the lifted relation T (R) ⊆ TX × TX ′ as follows:

T (R) := ((Tπ)(ρ), (Tπ′)(ρ)) | ρ ∈ TR,

where π : R → X and π′ : R → X ′ are the projection functions given byπ(x, x′) = x and π′(x, x′) = x′. In a diagram:

X Rπoo π′ // X ′

TX TRTπoo

〈Tπ,Tπ′〉

Tπ′ // TX ′

TR _

TX × TX ′

AA

]]:::::::::::::::::

In other words, we apply the functor T to the relation R, seen as a span

X Rπoo π′ // X ′ , and define TR as the image of TR under the product map

〈Tπ, Tπ′〉 obtained from the lifted projection maps Tπ and Tπ′.Let us first see some concrete examples.

5.2.5. Example. Fix a relation R ⊆ X × X ′. For the identity and constanctfunctors, we find, respectively:

IdR = R

CQR = ∆Q.

The relation lifting associated with the power set functor P can be definedconcretely as follows:

PR = (A,A′) ∈ PX × PX ′ | ∀a ∈ A ∃a′ ∈ A′.aRa′ and ∀a′ ∈ A′ ∃a ∈ A.aRa′.

This relation is known under many names, of which we mention that of theEgli-Milner lifting of R. For any standard, weak pullback preserving functor Tit can be shown that the lifting of Tω agrees with that of T , in the sense thatTωR = TR ∩ (TωX × TωX ′). From this it follows that

for all A ∈ TωX,A′ ∈ TωX ′ : A PωR A′ iff A PR A′,

and for this reason, we shall write PR rather than PωR.Relation lifting for the finitary multiset functor is slightly more involved: given

two maps µ ∈MωX,µ′ ∈MωX

′, we put µ MωR µ′ iff

there is some map ρ : R→ N such that ∀x ∈ X.∑ρ(x, x′) | x′ ∈ X ′ = 1

and ∀x′ ∈ X ′.∑ρ(x, x′) | x ∈ X = 1.


The definition of Dω is similar.Finally, relation lifting interacts well with various operations on functors [52].

In particular, we have

T0 T1R = T 0(T 1R)

T0 + T1R = T 0R ∪ T 1R

T0 × T1R =

((ξ0, ξ1), (ξ′0, ξ′1)) | (ξi, ξ′i) ∈ T i, for i ∈ 0, 1

.

TDR = (ϕ, ϕ′) | (ϕ(d), ϕ′(d) ∈ TR for all d ∈ D

5.2.6. Remark. Strictly speaking, the definition of the relation lifting of a givenrelation R depends on the type of the relation, i.e. given sets X,X ′, Y, Y ′ suchthat R ⊆ X ×X ′ and R ⊆ Y × Y ′, it matters whether we look at R as a relationfrom X to X ′ or as a relation from Y to Y ′. We have avoided this potential sourceof ambiguity by requiring the functor T to be standard, see also Fact 5.2.7(6).

Relation lifting has a number of properties that we will use throughout thechapter. It can be shown that relation lifting interacts well with the operation oftaking the graph of a function f : X → X ′, and with most operations on binaryrelations. Most of the properties below are easy to establish — we refer to [67] forproofs.

5.2.7. Fact. Let T be a set functor. Then the relation lifting T satisfies thefollowing properties, for all functions f : X → X ′, all relations R, S ⊆ X ×X ′,and all subsets Y ⊆ X, Y ′ ⊆ X ′:

1. T extends T : T (Grf) = Gr(Tf);

2. T preserves the diagonal: T (∆X) = ∆TX ;

3. T commutes with relation converse: T (R ) = (TR) ;

4. T is monotone: if R ⊆ S then T (R) ⊆ T (S);

5. T distributes over composition: T (R ; S) = T (R) ; T (S), if T preserves weakpullbacks.

6. T commutes with restriction: T (RY×Y ′) = TRTY×TY ′, if T is standardand preserves weak pullbacks.

Fact 5.2.7(5) plays a key role in our work. In fact, distributivity of T overrelation composition is equivalent to T preserving weak-pullbacks; the proof ofthis equivalence goes back to Trnkova [88].

Many proofs in this chapter will be based on Fact 5.2.7, and we will not alwaysprovide all technical details. In the lemma below we have isolated some facts thatwill be used a number of times; the proof may serve as a sample of an argumentusing properties of relation lifting.


5.2.8. Lemma. Let X, Y be sets, let f, g : X → Y be two functions and letR ⊆ X ×X and S ⊆ Y × Y be relations.

1. If (X,R) is a pre-order, then so is (TX, TR).

2. If f(x) S g(x) for all x ∈ X, then Tf(α) TS Tg(α) for all α ∈ TX.

3. If x R y implies f(x) S g(y) for all x, y ∈ X, then α TR β implies(Tf)α TS (Tg)β for all α, β ∈ TX.

Proof For part 1, observe that (X,R) is a pre-order iff ∆X ⊆ R and R ;R ⊆ R.Hence, if (X,R) is a pre-order, it follows from Fact 5.2.7(2,4) that ∆TX = T∆X ⊆TR, and from Fact 5.2.7(5,4) that TR ; TR = T (R ; R) ⊆ TR, implying that(TX, TR) is a pre-order as well.

For part 2, observe that the antecedent can be succinctly expressed as

(Grf ) ; Grg ⊆ S.

Then it follows by the properties of relation lifting that

(GrTf ) ; GrTg = (T (Grf)) ; T (Grg) (Fact 5.2.7(1))

= T ((Grf ) ) ; T (Grg) (Fact 5.2.7(3))

= T ((Grf ) ; Grg) (Fact 5.2.7(5))

⊆ TS (Fact 5.2.7(4))

But the inclusion (GrTf ) ; GrTg ⊆ TS is just another way of stating theconclusion of part 2.

For part 3, we reformulate the statement of its antecedent as

(Grf ) ;R ; Grg ⊆ S.

On the basis of this we may reason, via a completely analogous argument to theone just given, that

(GrTf ) ; TR ; GrTg ⊆ TS,

which is equivalent way of phrasing the conclusion of part 3.

Relation lifting interacts with the map Base as follows:

5.2.9. Fact ([67]). Let T be a standard, finitary, weak pullback-preserving func-tor.

1. Base is a natural transformation Base : T → Pω. That is, given a mapf : X → X ′ the following diagram commutes:

TXBaseX//

Tf

PωX

Pf

TX ′BaseX′// PωX

′


2. Given a relation R ⊆ X × X ′ and elements α ∈ TX, β ∈ TY , it followsfrom α TR β that Base(α) PR Base(β).

An interesting relation to which we shall apply relation lifting is the membershiprelation ∈. If needed, we will denote the membership relation restricted to a givenset X by the relation ∈X ⊆ X × PX. Given a set X and Φ ∈ TPX, we define

λTX(Φ) = α ∈ TX | α T∈X Φ.

Elements of λT (Φ) will be called lifted members of Φ.Properties of λT are intimately related to those of T .

5.2.10. Fact ([67]). The collection of maps λTX forms a distributive law withrespect to both the co- and the contravariant power set functor. That is, λT

provides two natural transformations, λT : TP → PT , and λT : T P → P T .

5.2.11. Remark. Rather than just a distributive law with respect to the functorP , λT is a distributive law over the monad (P, µ, η), in the sense of being alsocompatible with the unit η and the multiplication µ of P , as given by the followingtwo diagrams:

TX

ηTX $$HHHHHHHHHTηX // TPX

λTX

PTX

TPPX

TµX

λTPX // PTPXPλTX // PPTX

µX

TPXλTX

// PTX

In the terminology of [82], (T, λT ) is a monad opfunctor from the monad P toitself, and there is a one-one correspondence between the monad opfunctors andthe functors T equipped with extensions to endofunctors on the Kleisli categoryKl(M) associated with M . (The explicit results in [82], using the 2-functor AlgC,are in terms of monad functors and extensions to the category of Eilenberg-Moorealgebras. The results for monad opfunctors and the Kleisli category are dual.) Notethat the Kleisli category of the power set monad is (isomorphic to) the categoryRel with sets as objects, and binary relations as arrows. The correspondencementioned then links the natural transformation λT to the notion of relation liftingT .

5.2.12. Lemma. Let T be a standard, finitary, weak pullback-preserving functor.Let X be some set and let Φ ∈ TPX.

1. If ∅ ∈ Base(Φ) then λT (Φ) = ∅.

2. If Base(Φ) consists of singletons only, then λT (Φ) is a singleton.

3. If T maps finite sets to finite sets, then for all Φ ∈ TPωX, |λT (Φ)| < ω.


Proof For part 1, suppose that α is a lifted member of Φ; then it followsby Fact 5.2.9 that Base(α) P∈ Base(Φ). But from this it would follow, if∅ ∈ Base(Φ), that Base(α) contains a member of ∅, which is clearly impossible.Consequently, then λT (Φ) is empty.

For part 2, observe that another way of saying that Base(Φ) consists ofsingletons only, is that Φ ∈ TSX . Let θX : SX → X be the inverse of ηX , thatis, θX is the bijection mapping a singleton x to x. Clearly then, the mapTθX : TX → TSX is a bijection as well. In addition, we have (GrθX ) = ∈X , fromwhich it follows by Fact 5.2.7 that (GrTθX ) = T∈. From this it is immediatethat if Φ ∈ TSX , then (TθX)(Φ) is the unique lifted member of Φ.

Finally, we consider part 3. Since T is finitary, Φ ∈ TPωX implies thatΦ ∈ TPωY for some finite set Y , and from this it follows that Base(Φ) ⊆ PωY .If α is a lifted member of Φ, then by Fact 5.2.9 we obtain Base(α) P∈ Base(Φ),and so in particular we find Base(α) ⊆

⋃Base(Φ) ⊆ Y . From this it follows that

λT (Φ) ⊆ TY , and so by the assumption on T λT (Φ) must be finite.

5.2.4 Frames and their presentations

A frame is a complete lattice in which finite meets distribute over arbitrary joins.The signature of frames consists of arbitrary joins and finite meets, and it willbe convenient for us to include the top and bottom as well. Thus a frame willusually be given as L = 〈L,

∨,∧, 0, 1〉, while we will often consider join and meet

as functions∨

L : PL → L and∧

L : PωL → L. This enables us for instance todefine a frame homomorphism f : L → M as a map from L to M satisfyingf

∧=∧ (Pωf) and f

∨=∨ (Pf). By Fr we denote the category of

frames and frame homomorphisms. The initial frame (the lattice 0, 1 of truthvalues) will be denoted as Ω, and for a given frame L we will let !L denote theunique frame homomorphism from Ω to L, omitting the subscript if L is clearfrom context.

The order relation ≤L of a frame L is given by a ≤L b if a ∧ b = a (or,equivalently, a ∨ b = b). We can adjoin an implication operation to a frame L bydefining a→ b :=

∨c | a ∧ c ≤ b; this operation turns L into a Heyting algebra.

As a special case of implication we can consider the negation: ¬a :=∨c | a∧ c =

0. Neither of these two operations is preserved by frame homomorphisms. Asubset S of L is directed if for every s0, s1 ∈ S there is an element s ∈ S such thats0, s1 ≤ s. The join of a directed set S is often denoted as

∨↑ S.A frame presentation is a tuple 〈G | R〉 where G is a set of generators and

R ⊆ PPωG× PPωG is a set of relations. A presentation 〈G | R〉 presents a frameL if there exists a function f : G→ L which is compatible with R, i.e. such that

for all (t1, t2) ∈ R,∨A∈t1

∧(Pωf)A =

∨B∈t2

∧(Pωf)B,

and for all framesM and functions g : G→M compatible with R, there is a unique


frame homomorphism g′ : L→ M such that g′f = g. We call f the insertion ofgenerators (of G in L).

5.2.13. Fact ([91], Section 4.4). Every frame presentation presents a frame.

The details of the proof of the above fact tell us how to construct a unique framegiven a presentation 〈G | R〉. Omitting these details of the construction, we denotethis unique frame by Fr〈G | R〉. We will usually write

∨i∈I∧Ai =

∨j∈J∧Bj

instead of (Ai | i ∈ I, Bj | j ∈ J) when specifying relations. In light of the factthat a ≤ b iff a∨b = b, we will also allow ourselves the liberty to specify inequalitiesof the shape

∨i∈I∧Ai ≤

∨j∈J∧Bj as relations. It follows from the proof of Fact

5.2.13 that if f : G→ Fr〈G | R〉 is the insertion of generators, then every elementof Fr〈G | R〉 can be written as

∨i∈I∧PωfA for some Ai | i ∈ I ∈ PPωG; in

other words every element of Fr〈G | R〉 can be written as an infinite disjunctionof finite conjunctions of generators.

We will now introduce flat site presentations for frames, which have as oneof their main advantages that they allow us to assume that an arbitrary elementof the frame being presented is an infinite disjunction of generators. A flat siteis a triple 〈X,v, /0〉, where 〈X,v〉 is a pre-order and /0 ⊆ X × PX is a binaryrelation such that for all b v a /0 A, there exists B ⊆ ↓A ∩ ↓b such that b /0 B. Aflat site 〈X,v, /0〉 presents a frame L if there exists a function f : X → L suchthat

• f is order-preserving,

• 1 ≤∨

(Pf)X,

• for all a, b ∈ X, f(a) ∧ f(b) ≤∨

(Pf)(↓a ∩ ↓b), and

• for all a /0 A, f(a) ≤∨

(Pf)A

and for all frames M and all g : X →M satisfying the above two properties, thereexists a unique frame homomorphism g′ : L→M such that g′f = g. Specifically,for all a ∈ L,

g′(a) =∨g(x) | f(x) ≤ a.

To put it another way, the frame presented by a flat site is

Fr〈X,v, /0〉 ' Fr〈X | a ≤ b (a v b),a ≤

∨A (a /0 A),

1 =∨X

a ∧ b =∨c | c v a, c v b〉.

A suplattice is a complete∨

-semilattice; accordingly, a suplattice homomorphismis a map which preserves

∨. A suplattice presentation is a triple 〈X,v, /0〉 where

〈X,v〉 is a preorder and /0 ⊆ X × PX. A suplattice presentation 〈X,v, /0〉presents a suplattice L if there exists a function f : X → L such that


• f is order-preserving;

• for all a /0 A, f(a) ≤∨Pf(A);

and for all suplattices M and all functions g : X →M respecting the above twoconditions, there exists a unique suplattice homomorphism g′ : L→ M such thatg′ f = g. Every suplattice presentation presents a suplattice [60, Prop. 2.5].Now observe that every flat site can also be seen as a suplattice presentation withan additional stability condition. Consequently, given a flat site 〈X,v, /0〉, wecan generate two different objects with it: a frame Fr〈X,v, /0〉 and a suplatticeSupLat〈X,v, /0〉. The Flat site Coverage Theorem tells us that these two objectsare in fact order isomorphic.

5.2.14. Fact (Th. 5 of [93]). Let 〈X,v, /0〉 be a flat site. Then Fr〈X,v, /0〉 'SupLat〈X,v, /0〉.

We record the following consequences of the above fact. Suppose that 〈X,v, /0〉is a flat site which presents a frame L via f : X → L. Then

• every element of L is of the shape∨Pf(A) for some A ∈ PX;

• we can use 〈X,v, /0〉 both to define suplattice homomorphisms and framehomomorphisms.

5.2.5 Powerlocales via and ♦

We will now introduce the Vietoris powerlocale. In line with our generally algebraicapproach we shall define it directly as a functor on the category of frames ratherthan its opposite, the category of locales. In its full generality it originates (asthe “Vietoris construction”) in Johnstone [55], with some earlier, more restrictedreferences in [54]. For locales it is a localic analogue of hyperspace (with Vietoristopology). The points are (in bijection with) certain sublocales of the originallocale. For a full constructive description see [92].

Given a frame L, we first define L := L and L♦ := L, and then

V L := Fr〈L ⊕ L♦ | 1 = 1(a ∧ b) = a ∧b(∨↑A) =

∨↑a∈Aa (A ∈ PL directed)

♦(∨A) =

∨a∈A♦a (A ∈ PL)

a ∧ ♦b ≤ ♦(a ∧ b)(a ∨ b) ≤ a ∨ ♦b〉

5.2.15. Remark. We are abusing notation when specifying the relations in thedefinition above. Strictly speaking, we have two maps, : L → V L for theleft copy of L and ♦ : L♦ → V L for the right copy of L, so that the insertion ofgenerators is the map ⊕ ♦ : L ⊕ L♦ → V L.


Johnstone [55] shows that V gives a monad on the category of locales, i.e. acomonad on the category of frames. We shall not need the full strength of thishere, but some of the ingredients of the comonad structure are easy to check.

• V is functorial. If f : L→M is a frame homomorphism, then the function(f) ⊕ (♦f) : L ⊕ L♦ → VM is compatible with the relations in thepresentation of V L, so that there is a frame homomorphism V f : V L→ VMextending this map. It is also easy to show functoriality.

• The counit iL : V L→ L is given by a 7→ a and ♦a 7→ a. The comultiplica-tion µL : V L→ V V L is given by a 7→ a and ♦a 7→ ♦♦a.

There are various relations between properties of L and of V L. [55] showsthat L is regular, completely regular, zero-dimensional or compact regular iff V Lis, and also that if L is locally compact then so is V L. The same paper alsomentions without proof that if L is compact then so is V L, referring to a proofby transfinite induction similar to that used for the localic Tychonoff theorem in[54]. The paper leaves open the converse question, of whether V L being compactimplies that so is V L. We shall give here a constructive (topos-valid) proof usingpreframe techniques that L is compact iff V L is.

5.2.16. Definition. A frame L (or, more properly, its locale) is compact ifwhenever 1 ≤

∨↑i ai then 1 ≤ ai for some i.

The following constructive proof, as proposed by Steve Vickers, is a routineapplication of the techniques in [56].

5.2.17. Theorem. L is compact iff V L is.

Proof ⇒: L is compact iff the function L → Ω that maps a ∈ L to the truthvalue of a = 1 is a preframe homomorphism, i.e. preserves finite meets anddirected joins. This function is characterized by being right adjoint to the uniqueframe homomorphism ! : Ω→ L and so to prove compactness it suffices to definea preframe homomorphism L→ Ω and show that it is right adjoint to !. If L ispresented – as a frame – by generators and relations, then the “preframe coveragetheorem” of [56] shows how to derive a presentation as preframe, which can thenbe used for defining preframe homomorphisms from L. The strategy is to generatea ∨-semilattice from the generators, and add relations to ensure a ∨-stabilitycondition analogous to the ∧-stability used in Johnstone’s coverage theorem [54].


Our first step is to apply the preframe coverage theorem to derive a preframepresentation of V L. We show

V L ∼= Fr〈PωL× L (qua ∨ -semilattice) |1 ≤ (γ ∪ 1, d)(γ ∪ a, d) ∧ (γ ∪ b, d) ≤ (γ ∪ a ∧ b, d)

(γ ∪ ∨↑A, d) ≤

∨↑a∈A(γ ∪ a, d) (A directed)

(γ,∨↑A ∨ d) ≤

∨↑a∈A(γ, a ∨ d) (A directed)

(γ ∪ a, d) ∧ (γ, b ∨ d) ≤ (γ, (a ∧ b) ∨ d)(γ ∪ a ∨ b, d) ≤ (γ ∪ a, b ∨ d)〉.

The ∨-semilattice structure on PωL× L is the product structure from ∪ on PωLand ∨ on L. The homomorphisms between the frame presented above and V Lare given by

a 7→ (a, 0), ♦a 7→ (∅, a)

(γ, d) 7→∨c∈γ

c ∨ ♦d.

The relations shown are ∨-stable, so the preframe coverage shows that

V L ∼= PreFr〈PωL× L (qua poset) | same relations as above 〉.

We can now define a preframe homomorphism ϕ : V L→ Ω by

ϕ(γ, d) = ∃c ∈ γ. c ∨ d = 1.

To motivate this, we want criteria for∨c∈γ c∨♦d = 1, and intuitively this means

that for every sublocale K corresponding to a point of V L either K is includedin some c ∈ γ or K and d overlap. Taking K to be the closed complement of d,we get the given condition. This is not a rigorous argument, since that closedcomplement is not necessarily a point of V L. However, the rest of our argumentvalidates the choice. The relations in the preframe presentation of V L are largelyeasy to check. We shall just mention the penultimate one. Suppose (γ ∪ a, d)and (γ, b ∨ d) are both mapped to 1. We have either some c ∈ γ with c ∨ d = 1,in which case c∨ (a∧ b)∨ d = 1, or we have a∨ d = 1 and in addition some c′ ∈ γwith c′ ∨ b ∨ d = 1. In this latter case c′ ∨ (a ∧ b) ∨ d = 1.

Next we show that ϕ is right adjoint to ! : Ω → V L, the unique framehomomorphism defined by

!(p) =∨1 | p =

∨↑(0 ∪ 1 | p) .

We must show ϕ(!(p)) ≥ p and !(ϕ(γ, d)) ≤ (γ, d).

ϕ(!(p)) = ϕ(∨↑

(0 ∪ 1 | p))

= ϕ(∅, 0) ∨∨ϕ(1, 0) | p ≥ p

5.3. The T -powerlocale construction 157

since if p holds then the disjuncts include ϕ(1, 0) = 1. For the other inequality,we must show that ∨

1 | ϕ(γ, d) ≤ (γ, d).

If ϕ(γ, d) holds true then c ∨ d = 1 for some c ∈ γ, so

1 = (1, 0) = (c ∨ d, 0) ≤ (c, d) ≤ (γ, d).

⇐: Suppose in L we have 1 =∨↑i ai. Then in V L we have 1 = 1 =

∨↑i ai

and so 1 = ai for some i. Applying i to both sides gives 1 = ai.

5.3 The T -powerlocale construction

In this section we arrive at the main conceptual contribution of this chapter.Given a weak pullback-preserving, standard, finitary functor T : Set→ Set, wedefine its associated T -powerlocale functor VT : Fr→ Fr on the category of frames,using the Carioca axioms for coalgebraic modal logic. This construction trulygeneralizes the Vietoris powerlocale construction, because we will see that thePω-powerlocale is isomorphic to the Vietoris powerlocale. The other two majorresults in this section are the fact that one can lift a natural transformationbetween transition functors ρ : T ′ → T to a natural transformation ρ : VT → VT ′going in the other direction, and the fact that T -powerlocales are join-generated bytheir generators of the shape ∇α. We will establish the latter fact via the strongerresult by showing that VTL admits a flat site presentation. The fact that VTL isjoin-generated by its generators is not entirely surprising, since the Carioca axiomswere designed with the desirability of conjunction-free disjunctive normal forms inmind [17]; however the precise mathematical formulation of this property, usingflat sites and suplattices, is an improvement over what was previously known.

This section is organized as follows. In §5.3.1 we introduce the T -powerlocaleconstruction on frames. In §5.3.2 we make technical observations about T -powerlocales. In §5.3.3, we consider two instantiations of the T -powerlocaleconstruction, the most notable of which is the Pω-powerlocale which is isomorphicto the classical Vietoris powerlocale. In §5.3.4 we extend the T -powerlocale con-struction to a functor VT on the category of frames, and we show how one canlift natural transformations between set functors T , T ′ to natural transformationsbetween powerlocale functors VT , VT ′ . We conclude this section with §5.3.5, inwhich we show that the T -powerlocale construction admits a flat site presentation,a corollary of which is that each element of VTL has a disjunctive normal form.

5.3.1 Introducing the T -powerlocale

In this subsection, we will use the Carioca axioms for coalgebraic modal logic [17]to define the T -powerlocale VTL of a given frame L using a frame presentation,


i.e. using generators and relations. The generators of VTL will be given by theset TL; in order to specify the relations we will use relation lifting (§5.2.3) andslim redistributions, which we will introduce below. In addition, we will providean alternative presentation of VTL, which does not use slim redistributions. Froma conceptual viewpoint, it is not immediately obvious which presentation of VTLshould be taken as the primary definition. Our choice to use slim redistributionsin the primary definition is motivated by the existing literature [17, 66, 67].

5.3.1. Definition. Let X be a set and let Γ ∈ PωTX. The set of all slimredistributions of Γ is defined as follows:

SRD(Γ) =

Ψ ∈ TPω(⋃

γ∈ΓBase(γ))| ∀γ ∈ Γ, γ T∈ Ψ

Intuitively, Ψ ∈ TPωX is a slim redistribution of Γ ∈ PωTX if (i) Ψ is ‘obtained

from the material of Γ’, that is:

Ψ ∈ TPω(⋃

γ∈ΓBase(γ)),

and (ii) every element of Γ is a lifted member of Ψ, or equivalently, Γ ⊆ λT (Ψ).We illustrate this with the motivating example of slim redistributions, namelyslim redistribution for the finite powerset functor.

5.3.2. Example. Recall from Example 5.2.5 that if R ⊆ X × Y is a relationthen PωR ⊆ PωX × PωY can be characterized as follows:

αPωRβ iff ∀x ∈ α, ∃y ∈ β, xRy and ∀y ∈ β, ∃x ∈ α, xRy.

In particular, for ∈ ⊆ X × PX we get αPω ∈ Γ iff α ⊆⋃

Γ and ∀γ ∈ Γ, γ G α.(Recall that γ G α means that γ ∩ α is inhabited.) For an order ≤, let us definethe upper, lower and convex preorders on finite sets:

α ≤L β if α ⊆ ↓ β, i.e. ∀x ∈ α, ∃y ∈ β, x ≤ y

α ≤U β if ↑α ⊇ β, i.e. ∀y ∈ β, ∃x ∈ α, x ≤ y

α ≤C β if α ≤L β and α ≤U β.

Thus Pω ≤ is ≤C .Next, if α ∈ PωS then

Base(α) =⋂S ′ ∈ Pω(S) | α ⊆ S ′ = α.

From this, if Γ ∈ PωPωX then

SRD(Γ) = Ψ ∈ PωPω (⋃

Γ) | ∀γ ∈ Γ, (γ ⊆⋃

Ψ and ∀α ∈ Ψ, α G γ)= Ψ ∈ PωPω (X) |

⋃Ψ =

⋃Γ and ∀γ ∈ Γ, ∀α ∈ Ψ, α G γ.


5.3.3. Definition. Let T be a standard, finitary, weak pullback-preserving func-tor. Let L be a frame. We define the T -powerlocale of L

VTL := Fr〈TL | (∇1), (∇2), (∇3)〉,

where the relations are the Carioca axioms [17]:

(∇1) ∇α ≤ ∇β, (α T≤ β)(∇2)

∧α∈Γ∇α ≤

∨∇(T

∧)Ψ | Ψ ∈ SRD(Γ), (Γ ∈ PωTL)

(∇3) ∇(T∨

)Φ ≤∨∇β | β T∈ Φ, (Φ ∈ TPL)

5.3.4. Remark. To be precise, we assume that ∇ : TL→ VTL is the insertionof generators, so when specifying the relations we should write e.g. α ≤ β insteadof ∇α ≤ ∇β. The way we have specified the relations above is more consistentwith [17].

We will discuss the instantiation of these axioms for T = Pω in some more detailin §5.3.3.

We will now present a very useful equivalent definition of VTL. The crucialobservation behind the alternative definition of VTL is the following technicallemma, which characterizes the slim redistributions of a given finite subset Γ of〈TL, T≤〉 as the maximal lower bounds of Γ. Observe that the lemma also holdsin case Γ = ∅.

5.3.5. Lemma. Let L be a meet-semilattice (e.g., a frame) and let Γ ∈ PωTL.Then for any α ∈ TL, the following are equivalent:

(a) α ∈ TL is a lower bound of Γ, that is, α T≤ γ for all γ ∈ Γ;

(b) α T≤ (T∧

)Φ for some Φ ∈ SRD(Γ).

In particular, if Φ ∈ SRD(Γ) then (T∧

)Φ T≤ γ for all γ ∈ Γ.

Proof Recall that

SRD(Γ) :=

Ψ ∈ TP(⋃

γ∈ΓBase(γ))| Γ ⊆ λT (Ψ)

.

For the implication from (b) to (a), observe that for any a ∈ L and A ∈ PωL,we have that a ∈ A implies that

∧A ≤ a. By Fact 5.2.7 it follows that for all

γ ∈ TL and Ψ ∈ TPωL, if γ T∈ Ψ then T∧

(Ψ) T≤ γ. Now suppose that Ψ is aslim redistribution of Γ. Then Γ ⊆ λT (Ψ), and so (T

∧)Ψ is a T≤-lower bound of

Γ. From this the implication (b) ⇒ (a) is immediate.For the oppositie implication, take α ∈ TL such that ∀γ ∈ Γ, α T≤ γ.

Then by Fact 5.2.9, we obtain Base(α) P≤ Base(γ) for all γ ∈ Γ. AbbreviateC :=

⋃γ∈ΓBase(γ), and define f : Base(α)→ PC as follows:

f : a 7→ ↑L a ∩ C,


that is: f(a) = c ∈ C | a ≤ c. Then Tf is a function

Tf : TBase(α)→ TPC.

We claim that Ψ := Tf(α) is an element of SRD(Γ) and that α T≤ T∧

(Ψ). Forthe first claim, since Φ ∈ TPC, all we need to show is that Γ ⊆ λT (Ψ), i.e. thatfor all γ ∈ Γ, γ T∈ Ψ. So suppose that γ ∈ Γ; then by assumption, α T≤ γ, soBase(α) P≤ Base(γ). It follows from the definition of f that for all b ∈ Base(γ),and all a ∈ Base(α), if a ≤ b then b ∈ f(a). It follows by Fact 5.2.7 that

∀δ ∈ TBase(α), ∀β ∈ TBase(γ), δ T≤ β ⇒ β T∈ Tf(δ).

So in particular, since α ∈ TBase(α), γ ∈ TBase(γ) and α T≤ γ, we see thatγ T∈ Tf(α) = Ψ. Since γ ∈ Γ was arbitrary, it follows that Γ ⊆ λT (Ψ).Consequently, Ψ ∈ SRD(Γ), as we wanted to show.

For the second claim, i.e. that α T≤ T∧

(Ψ), it suffices to observe thata ≤

∧f(a) for all a ∈ Base(α), so by Fact 5.2.7,

∀δ ∈ TBase(α), δ T≤ T∧ Tf(δ).

Since α ∈ TBase(α) and Ψ = Tf(α), we get that α T≤ T∧ Tf(α) = T

∧(Ψ).

5.3.6. Corollary. Let L be a frame. Then

VTL ' Fr〈TL | (∇1), (∇2′), (∇3)〉,

where the relations are as follows:

(∇1) ∇α ≤ ∇β, (α T≤ β)(∇2′)

∧γ∈Γ∇γ ≤

∨∇α | ∀γ ∈ Γ, α T≤ γ, (Γ ∈ PωTL)

(∇3) ∇(T∨

)Φ ≤∨∇β | β T∈ Φ, (Φ ∈ TPL)

Proof Observe that the only difference between Fr〈TL | (∇1), (∇2′), (∇3)〉 andthe original definition of VTL is that we replaced (∇2),

(∇2)∧α∈Γ∇α ≤

∨∇(T

∧)Ψ | Ψ ∈ SRD(Γ), (Γ ∈ PωTL)

with (∇2′). The equivalence of these two relations is an immediate corollary ofLemma 5.3.5: take any Γ ∈ TPωL, then∨

∇T∧

(Ψ) | Ψ ∈ SRD(Γ)=∨∇α | ∃Ψ ∈ SRD(Γ), α T≤ ∇T

∧(Ψ) by order theory and (∇1),

=∨∇α | ∀γ ∈ Γ, α T≤ γ by Lemma 5.3.5.

It follows that VTL ' Fr〈TL | (∇1), (∇2′), (∇3)〉.

5.3.7. Remark. We will see later that both axioms (∇2) and (∇2′) are equallyuseful. It seems that (∇2′) has not been studied before in the literature oncoalgebraic modal logic via the ∇-modality [73, 17, 62, 67].


5.3.2 Basic properties of the T -powerlocale

In this subsection we make some technical observations about slim redistributionsand about the structure of the T -powerlocale. We start with two facts on slimredistributions.

5.3.8. Lemma. SRD(∅) = T∅.

Proof If Φ is a slim redistribution of the empty set, then by definition Φ ∈TPω(∅) = T∅. Conversely, any Φ ∈ T∅ satisfies the condition that ∅ ⊆λT (Φ), and so Φ ∈ SRD(∅).

The following Lemma plays an essential role when defining VT on framehomomorphisms, rather than just on frames. It is of crucial use when showingthat if f : L → M is a frame homomorphism, then VTf : VTL → VTM preservesconjunctions, as we will see in §5.3.4.

5.3.9. Lemma. Let X, Y be sets and let f : X → Y be a function; let Γ ∈ PωTX.Then the restriction of TPωf : TPωX → TPωY to SRD(Γ) is a surjection ontoSRD(PωTfΓ).

Proof Let X, Y, f and Γ be as in the statement of the Lemma, and abbreviateΓ′ := (PωTf)Γ, C :=

⋃γ∈Γ Base(γ) and C ′ :=

⋃γ′∈Γ′ Base(γ′). Then an easy

calculation shows that

C ′ =⋃γ∈Γ

Base(Tf)(γ) (definition of Γ′)

=⋃γ∈Γ

(Pf)Base(γ) (Base is natural transformation)

= (Pf)(C) (elementary set theory)

We will first show that TPωf maps slim redistributions of Γ to slim redistribu-tions of Γ′. For that purpose, take an arbitrary element Φ ∈ SRD(Γ), and writeΦ′ := (TPωf)Φ. We claim that Φ′ ∈ SRD(Γ′), and first show that

Φ′ ∈ TPωC ′, (5.1)

or equivalently, that BaseΦ′ ⊆ PωC′. To prove this inclusion, take an arbitrary

set A′ ∈ Base(Φ′). Since by Fact 5.2.9, Base(Φ′) = (PωPωf)(Base(Φ), this meansthat A′ must be of the form (Pωf)(A) for some A ∈ Base(Φ). In particular, A′

must be a subset of (Pωf)(⋃

Base(Φ)). Also, because Φ is a slim redistributionof Γ, by definition we have Base(Φ) ⊆ PωC, and so

⋃Base(Φ) ⊆

⋃C. From this

it follows that A′ ⊆ (Pf)(⋃

Base(Φ)) ⊆ (Pf)(⋃C) = C ′, as required.

Second, we claim thatΓ′ ⊆ λT (Φ′). (5.2)


To prove this, take an arbitrary element of Γ′, say, (Tf)γ for some γ ∈ Γ. Wehave γ T∈ Φ by the assumption that Φ ∈ SRD(Γ). But then, since a ∈ Aimplies fa ∈ (Pωf)A for any a ∈ C and A ⊆ C, it follows by Lemma 5.2.8 thatγ′ = (Tf)γ T∈ (TPωf)(Φ) = Φ′. This means that γ′ is a lifted member of Φ′, asrequired.

Clearly, the claims (5.1) and (5.2) above suffice to prove that Φ′ ∈ SRD(Γ′),which means that indeed, TPωf maps slim redistributions of Γ to slim redistribu-tions of Γ′.

Thus it is left to prove that every slim redistribution of Γ′ is of the form(TPωf)Φ for some slim redistribution Φ of Γ. Take an arbitrary Φ′ ∈ SRD(Γ′),and recall that P denotes the contravariant power set functor. Restrict f tothe map f− : C → C ′, which means that P f− : PωC

′ → PωC. It follows thatT Pf− : TPωC

′ → TPωC, so that we may define Φ := (T Pf−)Φ′, and obtainΦ ∈ TPωC. Hence, in order to prove that

Φ ∈ SRD(Γ), (5.3)

it suffices to show that Γ ⊆ λT (Φ). But this is an immediate consequence of thefact that λT is a distributive law of T over P (Fact 5.2.10), since for an arbitraryγ ∈ Γ we may reason as follows. From γ ∈ Γ it follows by definition of Γ′ that(Tf−)(γ) = (Tf)(γ) belongs to Γ′. Since Γ′ ⊆ λTY (Φ) by assumption, by definitionof P we find that γ ∈ (P Tf)λTY (Ψ). But by λT : T P → P T we know that(P Tf)λTY (Ψ) = λTX(T Pf)(Ψ) = λTX(Φ). Thus we find γ ∈ λT (Φ), as required.

Finally, observe that f− : C → C ′ is surjective, so that it follows by propertiesof the co- and contravariant power set functors that Pωf

− P f− = idPωC′ . Fromthis it is immediate by functoriality of T that

Φ′ = (TPωf− T Pf−)Φ′ = (TPωf

−)Φ = (TPωf)Φ.

This finishes the proof of the Lemma.

In the following lemma we gather some basic observations on the frame structureof the T -powerlocale. These facts generalize results from [67] to our geometricalsetting.

5.3.10. Lemma. Let T be a standard, finitary, weak pullback-preserving functorand let L be a frame.

1. If α ∈ TL is such that 0L ∈ Base(α), then ∇α = 0VTL.

2. If A ⊆ L is such that a ∧ b = 0L for all a 6= b in A, then ∇α ∧∇β = 0VTLfor all α 6= β in TA.

3. If there is no relation R such that α TR β, then ∇α ∧∇β = 0VTL.

4. 1VTL =∨∇γ | γ ∈ T1L.


5. For any A ⊆ L such that 1L =∨A, we have 1VTL =

∨∇α | α ∈ TA.

Proof For part 1, let α ∈ TL be such that 0L ∈ Base(α). Consider the mapf : L→ PL given by

f(a) :=

∅ if a = 0L,a if a > 0L.

Then idL =∨f , so that idTL = (T

∨) (Tf) by functoriality of T . In particular,

we obtain that α = (T∨

)(Tf)(α), so that we may calculate

∇α =∨∇β | β T∈ (Tf)(α)

(axiom ∇2)

≤∨∇β | Base(β) P∈ Base((Tf)(α))

(Fact 5.2.9)

=∨∅ (†)

= 0VTL

In order to justify the remaining step (†) in this calculation, observe that itfollows from the naturality of Base (Fact 5.2.9(1)) that

Base((Tf)(α)) = (Pf)(Base(α)),

and so by the assumption that 0L ∈ Base(α) we obtain ∅ ∈ Base((Tf)(α)). Nowsuppose for contradiction that there is some B ⊆ L such that B P∈ Base((Tf)(α)).Then by definition of P there is a b ∈ B such that b ∈ ∅, which provides thedesired contradiction. This proves (†), and finishes the proof of part 1.

For part 2, let A ⊆ L be such that a ∧ b = 0L for all a 6= b in A, and take twodistinct elements α, β ∈ TA. In order to prove that ∇α ∧∇β = 0VTL, it sufficesby axiom (∇2) to show that

∇(T∧

)(Φ) = 0VTL, for all Φ ∈ SRDα, β. (5.4)

Take an arbitrary slim redistribution Φ of α, β, then by Fact 5.2.12, Base(Φ)contains a set A0 ⊆ω A of size > 1. Define the map d : Base(Φ)→ Pω(A)∪1Lby putting:

d(B) :=

∅ if |B| > 1,B if |B| = 1,1L if |B| = 0.

It is straightforward to verify from the assumptions on A and the definition of d,that

∧B ≤

∨d(B), for each B ∈ Base(Φ). Hence it follows by Fact 5.2.7 that

(T∧

)(Φ) T≤ (T∨

)(Td)(Φ), so that by axiom (∇1) we may conclude that

∇(T∧

)(Φ) ≤ ∇(T∨

)(Td)(Φ) (5.5)

Finally, it follows from the naturality of Base (Fact 5.2.9(1)) that Base(Td)(Φ) =(Pd)(Base(Φ)). Consequently, for the set A0 ∈ Base(Φ) satisfying |A0| > 1, we


find ∅ = d(A0) ∈ Base(Td)(Φ), and then 0L =∨∅ ∈ (P

∨)Base(Td)(Φ) =

Base(T∨

)(Td)(Φ). Thus by part (1) of this lemma it follows that

∇(T∨

)(Td)(Φ) = 0VTL. (5.6)

This finishes the proof of part 2, since (5.4) is immediate on the basis of (5.5)and (5.6).

In order to prove part 3, suppose that α, β ∈ TL are not linked by any liftedrelation. Consider the (unique) map

f : L→ 1,

and define α′ := (Tf)α, β′ := (Tf)(β). Suppose for contradiction that α′ = β′.Then we would find α T ((Grf ) ; Grf) β, contradicting the assumption on α andβ. It follows that α′ and β are distinct, and so by part (2) of this lemma (withA = 1L), we may infer that ∇α′ ∧∇β′ = 0VTL. This means that we are done,since it follows from Grf ⊆ ≤ and the definitions of α′, β′, that α T≤ α′ andβ T≤ β′, and from this we obtain by (∇1) that

∇α ∧∇β ≤ ∇α′ ∧∇β′ ≤ 0VTL.

For part 4, we reason as follows:

1VTL =∨∇(T

∧)(Φ) | Φ ∈ SRD(∅), (axiom (∇2) with A = ∅)

=∨∇(T

∧)(Φ) | Φ ∈ T∅ (Fact 5.3.8)

=∨∇γ | γ ∈ T1L (‡)

where the last step (‡) is justified by the observation that, since the map∧

:PωL→ L restricts to a bijection

∧: ∅ → 1L, its lifting restricts to a bijection

T∧

: T∅ → T1L.

Finally, we turn to the proof of part 5. Let A ⊆ L be such that 1L =∨A, and

consider an arbitrary element Φ ∈ TA. We claim that

λT (Φ) ⊆ TA. (5.7)

To see this, take an arbitrary lifted element α of Φ. It follows from α T∈ Φthat Base(α) P∈ Base(Φ). In particular, each a ∈ Base(α) must belong to someB ∈ Base(Φ) ⊆ A. In other words, Base(α) ⊆ A, which is equivalent to sayingthat α ∈ TA. This proves (5.7).

By (5.7) and axiom (∇3) we obtain

∇(T∨

)(Φ) ≤∨∇α | α ∈ TA. (5.8)


Now we reason as follows:

1VTL =∨∇α | α ∈ T1L (part 4)

=∨∇(T

∨)(Φ) | Φ ∈ TA (∗)

≤∨∇α | α ∈ TA, (5.8)

To justify the second step (∗), observe that if we restrict the map∨

: PL→ L tothe bijection

∨: A → 1L, as its lifting we obtain a bijection T

∨: TA →

T1L.

5.3.3 Two examples of the T -powerlocale construction

In this subsection we will discuss two examples of T -powerlocales. First, wediscuss the somewhat trivial example of the Id-powerlocale. After that, we willdiscuss the defining example of T -powerlocales, namely the Pω-powerlocale, whichis isomorphic to the classical Vietoris powerlocale.

5.3.11. Example. Let Id: Set→ Set be the identity functor on the category ofsets. Then for all frames L, VIdL ' L.

First recall from Example 5.2.5 that for any relation R ⊆ X × Y , IdR = R.Moreover, if A ∈ IdPωL = PωL, then it is straightforward to verify that

SRD(A) = Ψ ∈ Pω(⋃c∈Ac) | ∀c ∈ A, c ∈ Ψ

= A.

Consequently, the ∇-relations reduce to the following in case T = Id:

(∇1) ∇a ≤ ∇b, (a ≤ b)(∇2)

∧a∈A∇a ≤ ∇

∧A, (A ∈ PωL)

(∇3) ∇∨A ≤

∨∇b | b ∈ A. (A ∈ PL)

The identity idL : L→ L obviously satisfies (∇1), (∇2) and (∇3). Moreover if wehave a frame M and a function f : L→M which is compatible with (∇1), (∇2)and (∇3), then it is easy to see that f is in fact a frame homomorphism L→M.By the universal property of frame presentations, it follows that VIdL ' L.

We now turn to the Pω-powerlocale. Recall from Example 5.2.3 that Pω : Set→Set, the covariant finite power set functor, is indeed standard, weak pullback-preserving and finitary. We will now show that the Pω-powerlocale is the Vietorispowerlocale. The equivalence of the ∇ axioms and the , ♦ axioms on distributivelattices is already known from the work of Palmigiano & Venema [73]; what isdifferent here is that we consider infinite joins rather than only finite joins.


We will use the presentation using (∇1), (∇2′) and (∇3) as our point ofdeparture. Recall that for all α, β ∈ PωL,

α ≤L β if α ⊆ ↓ β,α ≤U β if ↑α ⊇ β,

α ≤C β if α ≤L β and α ≤U β.

By Example 5.3.2, two of the relations presenting VPωL thus become

(∇2′)∧γ∈Γ∇γ ≤

∨∇α | ∀γ ∈ Γ, α ≤C γ

(∇3) ∇∨α | α ∈ Φ ≤

∨∇β | β ∈ Pω (

⋃Φ) and ∀α ∈ Φ, α G β

5.3.12. Lemma. We consider the presentation of VPωL.

1. In the presence of (∇1), the relation (∇2′) can be replaced by

(∇2.0) 1 ≤∨∇β | β ∈ PωL

(∇2.2) ∇γ1 ∧∇γ2 ≤∨∇β | β ≤C γ1, β ≤C γ2

2. In the presence of (∇1) and (∇2) (or its equivalent formulations), therelation (∇3) can be replaced by

(∇3.↑) ∇(γ ∪

∨↑ S) ≤ ∨↑ ∇ (γ ∪ a) | a ∈ S (S directed)

(∇3.0) ∇ (γ ∪ 0) ≤ 0(∇3.2) ∇ (γ ∪ a1 ∨ a2) ≤ ∇ (γ ∪ a1) ∨∇ (γ ∪ a2) ∨∇ (γ ∪ a1, a2)

Proof (1) (∇2.0) and (∇2.2) are special cases of (∇2′), when Γ is empty or adoubleton. To show that they imply (∇2′) is an induction on the number ofelements needed to enumerate the finite set Γ.

(2) Each of the replacement relations is a special case of (∇3) in which allexcept one of the elements of Φ are singletons. We now show that they aresufficient to imply (∇3). First, we show for any finite S that

∇(γ ∪

∨S)≤∨∇ (γ ∪ α) | ∅ 6= α ∈ PωS .

We use induction on the length of a finite enumeration of S. The base case, Sempty, is (∇3.0). Now suppose S = a ∪ S ′. Then

∇(γ ∪

∨S)

= ∇(γ ∪ a ∨

∨S ′)

≤ ∇ (γ ∪ a) ∨∇(γ ∪

∨S ′)∨∇

((γ ∪ a) ∪

∨S ′)

(by (∇3.2))

≤ ∇ (γ ∪ a) ∨∨∇ (γ ∪ α′) | ∅ 6= α′ ∈ PωS ′

∨∨∇ (γ ∪ a ∪ α′) | ∅ 6= α′ ∈ PωS ′ (by induction)

=∨∇ (γ ∪ α) | ∅ 6= α ∈ PωS .


Now we can use (∇3.↑) to relax the finiteness condition on S, since for an arbitraryS we have

∇(γ ∪

∨S)

= ∇(γ ∪

∨↑ ∨S0 | S0 ∈ PωS

)≤∨↑

∇(γ ∪

∨S0

)| S0 ∈ PωS

.

Finally, we can use induction on the length of a finite enumeration of Φ todeduce (∇3). More precisely, one shows by induction on n that

∇(γ ∪

∨S1, . . . ,

∨Sn)

≤∨

∇ (γ ∪ α) | ∅ 6= α ∈ Pω

(n⋃i=1

Si

)and ∀i, α G Si

.

5.3.13. Remark. Relation (∇2.0) can be weakened even further, to

1 ≤ ∇∅ ∨∇1.

For if β is non-empty then β ≤C 1. From (∇2.2) we can also deduce that∇∅ ∧∇1 = 0, giving that ∇∅ and ∇1 are clopen complements.

5.3.14. Lemma. In V L we have, for any S ⊆ L,

(∨

S)

=∨

(∨

α)∧∧a∈α

♦a | α ∈ PωS

.

Proof ≥ is immediate. For ≤, first note that since∨S is a directed join∨↑

α∈PωS∨α, we have (

∨S) ≤

∨↑α∈PωS (

∨α) and thus we reduce to the case

where S is finite. We show that for every α, β ∈ PωS we have

(∨

α ∨∨

β)∧∧a∈α

♦a ≤ RHS in statement,

after which the result follows by taking β = S and α = ∅. We use Pω-induction onβ, effectively an induction on the length of an enumeration of its elements. The


base case, β = ∅, is trivial. For the induction step, suppose β = β′ ∪ b. Then

(∨

α ∨∨

β)∧∧a∈α

♦a

= (∨

α ∨ b ∨∨

β′)∧∧a∈α

♦a

= (∨

α ∨ b ∨∨

β′)∧∧a∈α

♦a ∧((∨

α ∨∨

β′)∨ ♦b

)

=

((∨

α ∨∨

β′)∧∧a∈α

♦a

)∨

(∨ (α ∪ b) ∨∨

β′)∧

∧a∈α∪b

♦a

≤ RHS, by induction.

5.3.15. Theorem. Let L be a frame. Then V L ∼= VPωL.

Proof First we define a frame homomorphism ϕ : VPωL → V L by ϕ(∇α) = (∨α) ∧

∧a∈α♦a. We must check that this respects the relations. For (∇1),

suppose α ≤C β. From α ≤U β and α ≤L β we get∧a∈α♦a ≤

∧b∈β♦b and∨

α ≤∨β, giving ϕ(∇α) ≤ ϕ(∇β).

For (∇2.0), we have 1 = (0 ∨ 1) = 0 ∨ (1 ∧ ♦1) = ϕ (∇∅) ∨ ϕ (∇1).For (∇2.2), ϕ (∇γ1) ∧ ϕ (∇γ2) is

(∨γ1) ∧

∧c∈γ1♦c ∧ (

∨γ2) ∧

∧c′∈γ2♦c

′

= (∨γ1 ∧

∨γ2) ∧

∧c∈γ1♦c ∧

∧c′∈γ2♦c

′

= (∨γ1 ∧

∨γ2) ∧

∧c∈γ1♦ (c ∧

∨γ1 ∧

∨γ2) ∧

∧c′∈γ2♦ (c′ ∧

∨γ1 ∧

∨γ2)

= (∨γ1 ∧

∨γ2) ∧

∧c∈γ1♦ (c ∧

∨γ2) ∧

∧c′∈γ2♦ (c′ ∧

∨γ1)

= (∨

c∈γ1∨c′∈γ2c ∧ c

′)∧∧c∈γ1∨c′∈γ2♦ (c ∧ c′) ∧

∧c′∈γ2

∨c∈γ1♦ (c ∧ c′)

Redistributing the disjunctions of the ♦s, we find that each resulting disjunct isof the form

(∨


′)∧∧cRc′♦ (c ∧ c′)

for some R ∈ Pω (γ1 × γ2) such that γ1 PωRγ2. Note that for any such R if wedefine βR = c∧c′ | cRc′ then we have βR ≤C γi (i = 1, 2). Now by Lemma 5.3.14we see

(∨


′)∧∧cRc′♦ (c ∧ c′)

≤∨ (∨cR′c′c ∧ c

′) ∧∧c(R∪R′)c′♦ (c ∧ c′) | R′ ∈ Pω (γ1 × γ2)

≤∨ (

∨cR′c′c ∧ c

′) ∧∧cR′c′♦ (c ∧ c′) | R ⊆ R′ ∈ Pω (γ1 × γ2)

=∨ϕ (∇βR′) | R ⊆ R′ ∈ Pω (γ1 × γ2)


and the result follows.For (∇3.↑): the LHS is

(∨

γ ∨∨↑S) ∧∧c∈γ♦c ∧ ♦

(∨↑S)=∨↑ (

∨γ ∨ a) | a ∈ S ∧

∨↑ ∧c∈γ♦c ∧ ♦a | a ∈ S

=∨↑ (

∨γ ∨ a) ∧

∧c∈γ♦c ∧ ♦a | a ∈ S

which is the RHS.

For (∇3.0): the LHS is

(∨γ ∨ 0) ∧

∧c∈γ♦c ∧ ♦0 = 0.

For (∇3.2): the LHS is

(∨γ ∨ a1 ∨ a2) ∧

∧c∈γ♦c ∧ ♦ (a1 ∨ a2)

=2∨i=1

∨ (∨β) ∧

∧c∈β∪γ∪ai♦c | β ∈ Pω (γ ∪ a1, a2)

≤

2∨i=1

∨ϕ (∇ (β ∪ γ ∪ ai)) | β ∈ Pω (γ ∪ a1, a2)

=2∨i=1

∨ϕ (∇β) | γ ∪ ai ⊆ β ∈ Pω (γ ∪ a1, a2)

= ϕ (∇ (γ ∪ a1)) ∨ ϕ (∇ (γ ∪ a2)) ∨ ϕ (∇ (γ ∪ a1, a2)) .

Next, we define the frame homomorphism ψ : V L→ VPωL by

ψ (a) =∨∇α | α ≤L a = ∇∅ ∨∇a

ψ (♦a) =∨∇α | α ≤U a =

∨∇ (β ∪ a) | β ∈ PωL .

(Observe that the expression for ψ (♦a) could be simplified even further to∇1, a.)We check the relations. First, it is clear that ψ respects monotonicity of and ♦. preserves directed joins:

ψ((∨↑

i ai

))= ∇∅ ∨∇

∨↑i ai =

∨↑iψ (ai) .

preserves top immediately from (∇2.0). preserves binary meets:

ψ (a1) ∧ ψ (a2) = ∇∅ ∨ (∇a1 ∧ ∇a2)= ∇∅ ∨

∨∇β | β ≤C a1, β ≤C a2

= ∇∅ ∨∇a1 ∧ a2 = ψ ( (a1 ∧ a2)) .


♦ preserves joins:

ψ (♦ (∨A)) =

∨∇ (β ∪

∨A) | β ∈ PωL

=∨∇ (β ∪ α) | β ∈ PωL, ∅ 6= α ∈ PωA

=∨a∈A∨∇ (β ∪ a) | β ∈ PωL =

∨a∈Aψ(♦a).

For the first mixed relation, and noting that∇∅∧∇ (β ∪ b) ≤ ∇∅∧∇1 = 0,we have:

ψ (a) ∧ ψ (♦b) =∨β∈PωL (∇∅ ∨∇a) ∧∇ (β ∪ b)

=∨β∈PωL∇a ∧ ∇ (β ∪ b)

=∨∇γ | ∃β, γ ≤C a, γ ≤C β ∪ b

≤∨β∈PωL∇ (β ∪ a ∧ b) = ψ (♦ (a ∧ b)) .

For the second:

ψ ( (a ∨ b)) = ∇∅ ∨∇a ∨ b= ∇∅ ∨∇a ∨ ∇b ∨ ∇a, b≤ ψ (a) ∨ ψ (♦b)

since ∇∅ ∨∇a = ψ (a) and ∇b ∨ ∇a, b ≤ ψ (♦b).It remains to show that ϕ and ψ are mutually inverse.

ϕ (ψ (a)) = ϕ (∇∅ ∨∇a) = 0 ∨ (a ∧ ♦a) = a

since 0 ∧ ♦a ≤ ♦ (0 ∧ a) = 0.Next, to show ϕ (ψ (♦a)) = ♦a, we have

ϕ (ψ (♦a)) =∨β∈PωL

( (∨β ∨ a) ∧

∧b∈β♦b ∧ ♦a

)≤ ♦a= (1 ∨ a) ∧ ♦1 ∧ ♦a = ϕ (∇1, a) ≤ ϕ (ψ (♦a)) .

Finally, to show ψ (ϕ (∇α)) = ∇α, we have

ψ (ϕ (∇α)) = ψ( (∨α) ∧

∧a∈α♦a

)= (∇∅ ∨∇

∨α) ∧

∧a∈α∨βa∈PωL∇ (β ∪ a) .

Now,∧a∈α∨βa∈PωL∇ (β ∪ a) =

∨∇γ | ∀a ∈ α, ∃βa ∈ PωL, γ ≤C βa ∪ a

=∨∇γ | γ ≤U α .


Also

∇∅ ∧∨∇γ | γ ≤U α =

∨∇δ | δ ≤C ∅, δ ≤U α

=

∇α if α = ∅0 if α 6= ∅

∇∨α ∧

∨∇γ | γ ≤U α =

∨∇δ | δ ≤C

∨α , δ ≤U α

= ∇ (α ∪ ∨α)

=∨∇ (α ∪ α′) | ∅ 6= α′ ∈ Pωα

=

0 if α = ∅∇α if α 6= ∅

It follows that, whether α is empty or not, ψ (ϕ (∇α)) = ∇α.

5.3.4 Categorical properties of the T -powerlocale

In this section we discuss two categorical properties of the T -powerlocale construc-tion. First we show how to extend the frame construction VT to an endofunctoron the category Fr of frames. As a second topic we will see how the naturaltransformation i : VPω → VId (discussed in §5.2.5 as i : V → Id) can be generalizedto a natural transformation

ρ : VT → VT ′ ,

for any natural transformation ρ : T ′ → T satisfying some mild conditions (whereT and T ′ are two finitary, weak pullback preserving set functors).

VT is a functor

We start with introducing a natural way to transform a frame homomorphismf : L→M into a frame homomorphism from VTL to VTM. For that purpose wefirst prove the following technical lemma.

5.3.16. Lemma. Let L,M be frames and let f : L→M be a frame homomorphism.Then the map ∇ Tf : TL → VTM , i.e. α 7→ ∇(Tf)(α), is compatible with therelations (∇1), (∇2) and (∇3).

Proof We abbreviate ♥ := ∇ Tf , that is, for α ∈ TL, we define ♥α :=∇(Tf)(α).

In order to prove that ♥ is compatible with (∇1), we need to show that

for all α, β ∈ TL : α T≤L β implies ♥α ≤VTM ♥β. (5.9)

To see this, assume that α, β ∈ TL are such that α T≤L β. From this it followsby Lemma 5.2.8 and the assumption that f is a frame homomorphism, that(Tf)(α) T≤M (Tf)(β). Then by (∇1)M we obtain that ♥α ≤VTM ♥β, as required.


Proving compatibility with (∇2) boils down to showing

for all Γ ∈ PωTL :∧α∈Γ

♥α ≤∨♥(T

∧)(Ψ) | Ψ ∈ SRD(Γ). (5.10)

For this purpose, given Γ ∈ PωTL, let Γ′ ∈ PωTM denote the set (PωTf)(Γ) =(Tf)(α) | α ∈ Γ. Then we may observe∧

α∈Γ

♥α =∨∇(T

∧)(Ψ) | Ψ ∈ SRD(Γ′) (∇1)

≤∨∇(T

∧)(TPωf)(Φ) | Φ ∈ SRD(Γ) (Lemma 5.3.9)

=∨∇(Tf)(T

∧)(Φ) | Φ ∈ SRD(Γ) (†)

=∨♥(T

∧)(Φ) | Φ ∈ SRD(Γ) (definition of ♥)

Here the identity marked (†) is easily justified by f being a homomorphism:it follows from f

∧=∧(Pωf) and functoriality of T that (Tf) (T

∧) =

(T∧

) (TPωf).Finally, for compatibility with (∇3) we need to verify that

for all Φ ∈ TPL : ♥(T∨

)(Ψ) ≤∨♥β | β T∈ Φ. (5.11)

To prove this, we calculate for a given Φ ∈ TPL:

♥(T∨

)(Φ) = ∇(Tf)(T∨

)(Φ) (definition of ♥)

= ∇(T∨

)(TPf)(Φ) (f a frame homomorphism)

≤∨∇β | β T∈ (TPf)(Φ) (∇3)M

=∨∇(Tf)(γ) | γ T∈ Φ (‡)

=∨♥γ | γ T∈ Φ (definition of ♥)

Here the identity (‡) follows from the observation that for all β ∈ TM andΦ ∈ TPL, we have β T∈ (TPf)(Φ) iff β is of the form β = (Tf)(γ) for someγ ∈ TL. Using Fact 5.2.7, this is easily derived from the observation that forb ∈M and A ∈ PL, we have b ∈ (Pf)A iff b = f(c) for some c ∈ A.

Lemma 5.3.16 justifies the following definition.

5.3.17. Definition. Let f : L → M be a frame homomorphism. We defineVTf : VTL→ VTM to be the unique frame homomorphism extending

∇ Tf : TL→ VTM.


5.3.18. Theorem. Let T be a standard, finitary, weak pullback-preserving func-tor. Then the operation VT defined above is an endofunctor on the categoryFr.

Proof Since for an arbitrary f : L→M we have ensured by definition that VTFis a frame homomorphism from VTL to VTM, it is left to show that VT maps theidentity map of a frame to the identity map of its T-powerlocale, and distributesover function composition. We confine our attention to the second property.

Let f : K → L and g : L → M be two frame homomorphisms. In order toshow that VT (g f) = VTg VTf , first recall that VT (g f) is by definition theunique frame homomorphism extending the map ∇M T (g f) : TK → VTM.Hence, it suffices to prove that the map VTg VTf , which is obviously a framehomomorphism, extends ∇M T (g f). But it is easy to see why this is the case:given an arbitrary element α ∈ TK, a straightforward unravelling of definitionsshows that

(VTg VTf)(α) = VTg(∇L(Tf)(α)) = ∇M(Tg)(Tf)(α) = ∇MT (g f)(α),

as required.

Natural transformations between VT and VT ′

Now that we have seen how each finitary, weak pulback preserving set functor Tinduces a functor VT on the category of frames, we investigate the relation betweentwo such functors VT , VT ′ . In fact, we have already seen an example of this: recallthat in §5.2.5 we mentioned Johnstone’s result [55] that the standard Vietorisfunctor V is in fact a comonad on the category of frames. In our nabla-basedpresentation of this functor as V = VPω , thinking of the identity functor on thecategory Fr as the Vietoris functor VId , we can see the counit of this comonad asa natural transformation

i : VPω → VId ,

given by iL : ∇A 7→∧A. More precisely, we can show that the map ♥ : PωL→ L

given by ♥A :=∧A is compatible with the ∇-axioms, and hence can be uniquely

extended to the homomorphism iL; subsequently we can show that this i is naturalin L. Recall that in the case of a concrete topological space (X, τ), this counitcorresponds on the dual side to the singleton map σX : s 7→ s which providesan embedding of a compact Hausdorff topology into its Vietoris space.

We will now see how to generalize this picture, of the natural transformationi : VPω → VId being induced by the singleton natural transformation σ : Id → Pω,to a more general setting. First consider the following definition.

5.3.19. Definition. Let T and T ′ be standard, finitary, weak pullback-preservingfunctors. A natural transformation ρ : T ′ → T is said to respect relation lifting if


for any relation R ⊆ X × Y we have, for all α, β ∈ TX

α TR β only if ρX(α) T ′R ρY (β). (5.12)

We call ρ base-invariant if it commutes with Base, that is,

BaseT′= BaseT ρ. (5.13)

for any set X.

5.3.20. Example. We record three examples of base-invariant natural transfor-mations which respect relation lifting.

1. The base transformation BaseT : T → Pω;

2. The singleton natural transformation σ : Id→ Pω, which is in fact a specialcase of (1);

3. The diagonal map δ (given by δX : x 7→ (x, x)); it is straightforward tocheck that as a natural transformation, δ : Id → Id × Id also satisfies bothproperties of Definition 5.3.19.

As we will see now, every base-invariant natural transformation ρ : T ′ → Tthat respects relation lifting, induces a natural transformation ρ : VT → VT ′ . Inparticular, the natural transformation i : V → Id can be seen as i = σ, whereσ : Id → Pω is the singleton transformation discussed above.

5.3.21. Theorem. Let T and T ′ be standard, finitary, weak pullback-preservingfunctors, assume that ρ : T ′ → T is a base-invariant natural transformation thatrespects relation lifting, and let L be a frame. Then the map from TL to VT ′Lgiven by

α 7→∨∇α′ | α′ ∈ T ′L, ρ(α′) T≤ α

specifies a frame homomorphism

ρL : VTL→ VT ′L

which is natural in L.

Proof We let ♥ : TL→ L denote the map given in the statement of the Theorem,that is, ♥α :=

∨∇α′ | α′ ∈ T ′L, ρ(α′) T≤ α. We will first prove that this

map is compatible with, respectively, (∇1), (∇2) and (∇3), and then turn to thenaturality of the induced frame homomorphism.

1. Claim. The map ♥ is compatible with (∇1).


Proof of Claim To show that ♥ is compatible with (∇1), take two elementsα, β ∈ TL such that α T≤ β. Then for any α′ ∈ T ′L such that ρ(α′) T≤ α, bytransitivity of T≤ (Fact 5.2.7(5)), we obtain that ρ(α′) T≤ β. From this it isimmediate that ♥α ≤ ♥β, as required.


Proof of Claim For compatibility with (∇2), it suffices to show compatibility with(∇2′). That is, for Γ ∈ PωTL, we will verify that∧

♥γ | γ ∈ Γ ≤∨♥β | β T≤ γ, for all γ ∈ Γ. (5.14)

We start with rewriting the left hand side of (5.14) into∧♥γ | γ ∈ Γ =

∧∨∇γ′ | ρ(γ′) T≤ γ

| γ ∈ Γ

(definition of ♥)

=∨∧

ϕγ | γ ∈ Γ | ϕ ∈ CΓ

(frame distributivity)

where we define CΓ := ϕ : Γ→ T ′L | ρ(ϕγ) T≤ γ, for all γ ∈ Γ.For any map ϕ ∈ CΓ we may calculate∧ϕγ | γ ∈ Γ

=∨∇γ′ | γ′ T ′≤ ϕγ,∀γ ∈ Γ (∇2′)

≤∨∇γ′ | ρ(γ′) T≤ ρ(ϕγ),∀γ ∈ Γ (ρ respects relation lifting)

≤∨∇γ′ | ρ(γ′) T≤ γ, ∀γ ∈ Γ (ϕ ∈ CΓ, transitivity of T≤)

=∨∨

∇γ′ | ρ(γ′) T≤ β | β T≤ γ, ∀γ ∈ Γ

(associativity of∨

)

=∨♥β | β T≤ γ, ∀γ ∈ Γ (definition of ♥)

From the above calculations, (5.14) is immediate.


Proof of Claim We need to show, for an arbitrary but fixed set Φ ∈ TPL, that

♥(T∨

)(Φ) =∨♥α | α T∈ Φ. (5.15)

By definition, on the left hand side of (5.15) we find

♥(T∨

)(Φ) =∨∇β′ | ρ(β′) T≤ (T

∨)(Φ),


while on the right hand side we obtain, by definition of ♥,∨♥α | α T∈ Φ =

∨∨∇α′ | ρ(α′) T≤ α | α T∈ Φ

=∨∇α′ | ρ(α′) T (≤ ; ∈) Φ

where the latter equality is by associativity of

∨, and the compositionality of

relation lifting (Fact 5.2.7(5)).As a consequence, in order to establish the compatibility of ♥ with (∇3), it

suffices to show that

∇β′ ≤∨∇α′ | ρ(α′) T (≤ ; ∈) Φ

for any β′ with ρ(β′) T≤ (T

∨)(Φ). (5.16)

Let β′ be an arbitrary element of TL such that ρ(β′) T≤ (T∨

)(Φ). Our goal willbe to find a set Φ′ ∈ T ′PL satisfying (5.20), (5.21) and (5.22) below: clearly thiswill satisfy to prove (5.16).

By Fact 5.2.9 we obtain that

BaseT (ρβ) P≤ BaseT ((T∨

)(Φ)) = (P∨

)BaseT (Φ),

and since ρ is base-invariant, we have BaseT′(β′) = BaseT (ρβ′). Combining these

facts we see that BaseT′(β′) P≤ (P

∨)BaseT (Φ). This motivates the definition of

the following map H : BaseT′(β′)→ PωPL:

H(b) := B ∈ BaseT (Φ) | b ≤∨B.

From the definitions it is immediate that

for all b ∈ BaseT′(β′) : b ≤

∧∨B | B ∈ H(b)

. (5.17)

Also, given a set B ∈ PωPL, let CB be the collection of choice functions on B, thatis:

CB := f : B → L | f(B) ∈ B for all B ∈ B.

Then it follows by frame distributivity that∧∨B | B ∈ B

=∨∧

(Pf)(B) | f ∈ CB. (5.18)

Define the map K : PωPL→ PL by

K(B) :=∧

(Pf)(B) | f ∈ CB,

then it follows from (5.17), (5.18) and the definitions that

for all b ∈ BaseT′(β′) : b ≤

∨K(H(b)). (5.19)


As a corollary, if we define

Φ′ := (T ′K)(T ′H)(β′),

then it follows from (5.19), by the properties of relation lifting, that β′ T ′≤(T ′∨

)(Φ′), so that an application of (∇1) yields

∇β′ ≤ ∇(T ′∨

)(Φ′). (5.20)

Also, on the basis of an application of (∇3) we may conclude that

∇(T ′∨

)(Φ′) ≤∨∇γ′ | γ′ T ′∈ Φ′. (5.21)

This means that we are done with the proof of (5.16) if we can show that

for any γ′ ∈ T ′L, if γ′ T ′∈ Φ′ then ρ(γ′) T (≤ ; ∈) Φ. (5.22)

For a proof of (5.22), let γ′ be an arbitrary T ′-lifted member of Φ′ and recallthat Φ′ = (TK)(TH)(β′). Then it follows by the assumption that ρ respectsrelation lifting, that ρ(γ′) T∈ ρ(Φ′) = (TK)(TH)(ρ(β′)). Given our assumptionon β′, this means that the relation between ρ(γ′) and Φ can be summarized as

ρ(γ′) T∈ (TK)(TH)(β) and β T≤ (T∨

)(Φ) for some β ∈ TBaseT′(β′), (5.23)

where for β we may take ρ(β′).Returning to the ground level, observe that for any c ∈ L, A ∈ BaseT (Φ), we

have

if c ∈ KH(b) and b ≤∨A, for some b ∈ BaseT

′(β′), then c (≤ ; ∈) A. (5.24)

To see why this is the case, assume that c ∈ KH(b) and b ≤∨A, for some

b ∈ BaseT′(β′). Then by definition of H we find A ∈ H(b), while c ∈ KH(b)

simply means that c =∧f(B) | B ∈ H(b), for some f ∈ CH(b). But then it is

immediate that c ≤ f(A), while f(A) ∈ A by definition of CH(b). Thus f(A) is therequired element witnessing that c (≤ ; ∈) A.

But by the properties of relation lifting, we may derive from (5.24) that

if γ T∈ (TK)(TH)(β) and β T≤ (T∨

)(Φ) for some β ∈ TBaseT′(β′),

then γ T (≤ ; ∈) Φ, (5.25)

so that it is immediate by (5.23) that ρ(γ′) T (≤ ; ∈) Φ. This proves (5.22).As mentioned already, the compatibility of ♥ with (∇3) is immediate by (5.20),

(5.21) and (5.22), and so this finishes the proof of Claim 3.

As a corollary of the Claims 1–3, we may uniquely extend♥ to a homomorphismρL : VTL→ VT ′L. Clearly then, in order to prove the theorem it suffices to provethe following claim.


4. Claim. The family of homomorphisms ρL constitutes a natural transformationρ : VT → VT ′ .

Proof of Claim Given two frames L and M and a frame homomorphism f : L→M,we need to show that the following diagram commutes:

VTLbρL //

VT f

VT ′LVT ′f

VTM

bρM // VT ′MTo show this, take an arbitrary element α ∈ TL, and consider the followingcalculation:

(VT ′f)(ρL(∇α))

= (VT ′f)(♥α) (definition of ρL))

= (VT ′f)(∨

∇β′ | ρL(β′) T≤ α)

(definition of ♥))

=∨

(V ′Tf)(∇β′) | ρL(β′) T≤ α

(VT ′f is a frame homomorphism)

=∨∇(T ′f)(β′) | ρL(β′) T≤ α

(definition of VT ′f))

=∨∇δ′ | ρM(δ′) T≤ (Tf)(α)

(†)

= ♥(Tf)(α) (definition of ♥))

= ρM(∇(Tf)(α)) (definition of ρM))

= ρM((VTf)(∇α)) (definition of VTf))

Here the crucial step, marked (†), is proved by establishing the two respectiveinequalities, as follows. For the inequality ≤, it is straightforward to show thatthe set of joinands on the left hand side is included in that on the right hand side,and this follows from

ρL(β′) T≤ α implies ρM((T ′f)(β′)) T≤ (Tf)(α). (5.26)

To prove (5.26), suppose that ρL(β′) T≤ α; then it follows by the fact that f is ahomomorphism, and hence, monotone, that (Tf)(ρL(β′)) T≤ (Tf)(α). But sinceρ is a natural transformation, we also have (Tf)(ρL(β′)) = ρM (T ′f)(β′), and fromthis (5.26) is immediate.

In order to prove the opposite inequality∨∇δ′ | ρM(δ′) T≤ (Tf)(α)

≤∨∇(T ′f)(β′) | ρL(β′) T≤ α

, (5.27)

fix an arbitrary element δ′ ∈ TL such that ρM(δ′) T≤ (Tf)(α).


Define the map h : BaseT′(δ′)→ L by putting

h(d) :=∧a ∈ BaseT (α) | d ≤ f(a).

Then for all d ∈ BaseT′(δ′) and all a ∈ BaseT (α), we find that d ≤ fa implies

hd ≤ a; this can be expressed by the relational inclusion

Grf ;≥ ; Grh ⊆ ≥

so that by the properties of relation lifting we may conclude that Gr(Tf) ; T≥ ;Gr(Th) ⊆ T≥, which is just another way of saying that, for all δ ∈ TBaseT

′(δ′),

we haveδ T≤ (Tf)(α) only if (Th)(δ) T≤ α. (5.28)

Now defineβ′ := (T ′h)(δ′),

then we may conclude from the fact that ρ respects relation lifting that ρL(β′) =(Th)ρM(δ′), and so by the assumption that ρM(δ′) T≤ (Tf)(α), we obtain by(5.28) that

ρL(β′) T≤ α. (5.29)

Similarly, from the fact that d ≤ fhd, for each d ∈ BaseT′(δ′), we may derive

that δ′ T ′≤ (T ′f)(β′), and so by (∇1) we may conclude that

∇δ′ ≤ ∇(T ′f)(β′). (5.30)

Finally, (5.26) is immediate by (5.25) and (5.30).This finishes the proof of Claim 4.

5.3.22. Remark. The definition of the ρL : VTL→ VT ′L, using the assignment

α 7→∨∇α′ | α′ ∈ T ′L, ρ(α′) T≤ α,

is very similar to that of a right adjoint. If it were the case that ρL preserved allmeets, then the adjoint functor theorem would allow us to define its left adjoint.However, we only have a proof that ρL : VTL→ VT ′L preserves finite conjunctions,so it is not at all obvious at this point if there even is a left adjoint to ρL. This isan interesting question for future work.

5.3.5 T -powerlocales via flat sites

In this subsection, we will show that VTL, the T -powerlocale of a given frame L,has a flat site presentation as VTL ' Fr〈TL, T≤, /L

0 〉. It then follows by the Flatsite Coverage Theorem that every element of VTL has a disjunctive normal form,


and that the suplattice reduct of VTL has a presentation defined only in terms ofthe order T≤ and the lifted join function T

∨: TPL→ TL.

Recall that 〈X,v, /0〉 is a flat site if 〈X,v〉 is a pre-order and /0 is a basiccover relation compatible with v. In that case, we know that 〈X,v, /0〉 presents aframe Fr〈X,v, /0〉, and that if we denote the insertion of generators by ♥ : X →Fr〈X,v, /0〉, then

Fr〈X,v, /0〉 ' Fr〈X | ♥a ≤ ♥b (a v b),1 =

∨♥a | a ∈ X

♥a ∧ ♥b =∨♥c | c v a, c v b

♥a ≤∨♥b | b ∈ A (a /0 A)〉.

Observe that this is very similar to our presentation of VTL from Corollary 5.3.6using (∇1), (∇2′) and (∇3), namely

VTL ' Fr〈TL | ∇α ≤ ∇β (α T≤ β),∧Γ∇γ =

∨∇δ | ∀γ ∈ Γ, δ T≤ γ (Γ ∈ TPωL)

∇T∨

(Φ) ≤∨∇β | β ∈ λT (Φ) (Φ ∈ TPL)〉.

We will see below that if we define a cover relation /L0 which is inspired by (∇3),

then we obtain a flat site 〈TL, T≤, /L0 〉, and this flat site presents VTL.

So how do we go about defining a basic cover relation /L0 ⊆ TL× PTL so we

can give a presentation of VTL? Intuitively, we would like to take the T -lifting ofthe relation (a,A) ∈ L× PL | a ≤

∨A = ≤ ; (Gr

∨) . However, the T -lifting of

this relation is of type TL×TPL, while a basic cover relation on 〈TL, T≤〉 shouldbe of type TL × PTL. We solve this by involving the natural transformationλT : TP → PT , given by λT (Φ) := β ∈ TL | β T∈ Φ, assigning to eachΦ ∈ TPL the set of its lifted members. That is, we define

/L0 := (α, λT (Φ)) ∈ L× PTL | α T≤ T

∨(Φ).

In other words: we put α /L0 Γ iff Γ is of the form λT (Φ) for some Φ ∈ TPL

such that α T≤ (T∨

)Φ. We must now do two things: first, we must show that〈TL, T≤, /L

0 〉 is a flat site, meaning that /0 is compatible with T≤. Secondly, wemust show that 〈TL, T≤, /L

0 〉 presents VTL. The following technical observationabout the relation α T≤ T

∨(Φ) is the main reason why VTL admits a flat site

presentation. The reason for introducing a ∧-semilattice M below will becomeapparent in §5.4.2.

5.3.23. Lemma. Let L be a frame and let M be a ∧-subsemilattice of L. Thenfor all α ∈ TM and Φ ∈ TPM such that α T≤ T

∨(Φ), there exists Φ′ ∈ TPM

such that

1. α T≤ T∨

(Φ′);


2. Φ′ T⊆ T ↓L Tη(α);

3. Φ′ T⊆ T ↓L(Φ)

Proof First, we define the following relation on M × PM :

R := (a,A) ∈M × PM | a ≤∨A = (≤ ; (Gr

∨) ) M×PM .

Consider the span Mp1←− R

p2−→ PM . We define the following function f : R→ R:

f : (a,A) 7→ (a, a ∧ A),

where a ∧ A := a ∧ b | b ∈ A. To see why this function is well-defined, firstobserve that a∧A ∈ PM because M is a ∧-subsemilattice of L. Moreover, by theinfinite distributive law for frames, we see that if (a,A) ∈ R, i.e. if a ≤

∨A, then

also a ≤∨

(a ∧A), so that (a, a ∧A) ∈ R. Now observe that f : R→ R satisfiesan equation and two inequations: for all (a,A) ∈ R,

p1 f(a,A) = a = p1(a,A), by def. of f ,

p2 f(a,A) = a ∧ A ⊆L ↓La = ↓L ηL p1(a,A), since ∀b ∈ A, a ∧ b ≤ a,

p2 f(a,A) = a ∧ A ⊆L ↓LA = ↓L p2(a,A) since ∀b ∈ A, a ∧ b ≤ b ∈ A.

Now consider the lifted diagram

TM TRTp1oo Tp2 // TPM.

It follows from Lemma 5.2.8 and the equation/inequations above that for eachδ ∈ TR, we have

Tp1 Tf(δ) = Tp1(δ), (5.31)

Tp2 Tf(δ) T⊆L T ↓L TηL Tp1(δ), (5.32)

Tp2 Tf(δ) T⊆L T ↓L Tp2(δ) (5.33)

Now recall that by Fact 5.2.7,

T≤ ; Gr(T∨

) = T (≤ ; (Gr∨

) ) = TR,

so we see that α T ≤ T∨

(Φ) iff α TR Φ. So let α ∈ TM and Φ ∈ TPM suchthat α T≤ T

∨(Φ), i.e. such that α TR Φ; we will show that there is a Φ′ ∈ TPM

satisfying properties (1)–(3). First, observe that by definition of relation lifting,there must exist some δ ∈ TR such that

Tp1(δ) = α and Tp2(δ) = Φ.


We claim that Φ′ := Tp2Tf(δ) satisfies properties (1)–(3). We know by definitionof relation lifting that (Tp1 Tf(δ)) TR (Tp2 Tf(δ)). Since

Tp1 Tf(δ) = Tp1(δ) by (5.31),

= α by assumption,

it follows that α TR Φ′, i.e. α T≤ T∨

(Φ′); we conclude that (1) holds. Moreover,it follows immediately from (5.32) that (2) holds. Similarly, it follows immediatelyfrom (5.33) that (3) holds.

In the lemma above, we use the lifted inclusion relation T⊆ and the lifteddownset function T ↓. In the lemma below we record several elementary obser-vations about the interaction between T⊆, T ↓ and the natural transformationλT : TP → PT .

5.3.24. Lemma. Let 〈X,v〉 be a pre-order, let α ∈ TX and let Φ,Φ′ ∈ TPX.Then

1. ↓TXλT (Φ) = λT (T↓X(Φ));

2. ↓TXα = λT (T↓X TηX(α));

3. If Φ′ T ⊆X Φ, then also λT (Φ′) ⊆ λT (Φ).

Proof (1). For all a ∈ X and all A ∈ PX, we have a ≤ ; ∈ A iff a ∈ ↓X A.Consequently,

∀α ∈ TL,∀Φ ∈ TPL, α T≤ ; T∈ Φ iff α T∈ T ↓X(Φ).

Now we see that

α ∈ ↓TX λT (Φ)⇔ α T≤ ; T∈ Φ by def. of ↓ and λT ,

⇔ α T∈ T ↓X(Φ) by the above,

⇔ α ∈ λT (T↓X(Φ)) by def. of λT .

(2). For all a, b ∈ X, we have b ≤ a iff b ∈ ↓Xa. It follows by relation liftingthat

∀α, β ∈ TX, β T≤ α iff β T∈ T ↓X TηX(α).

It now follows by an argument analogous to that for (1) above that (2) holds.(3). Observe that for all A,A′ ∈ PX and all a ∈ X, we have that a ∈ A′ ⊆ A

implies that a ∈ A. The statement follows by relation lifting.

We are now ready to prove that 〈TL, T≤, /L0 〉 is indeed a flat site.

5.3.25. Lemma. If L is a frame then 〈TL, T≤, /L0 〉 is a flat site.


Proof We have already know from Lemma 5.2.8 that 〈TL, T≤〉 is a pre-order, sowhat remains to be shown is that the relation /L

0 is compatible with the pre-order.Fix α ∈ TL and Φ ∈ TPL such that α T≤ T

∨(Φ), so that α /L

0 λT (Φ). We need

to show that

∀β ∈ TL, if β T≤ α then ∃Γ ∈ TPL s.t. Γ ⊆ ↓TLβ ∩ ↓TL λT (Φ) and β /L0 Γ.

(5.34)But this is easy to see: if β T≤ α then since α T≤ T

∨(Φ), it follows by transitivity

of T≤ that β T≤ T∨

(Φ). Now by Lemma 5.3.23 there exists Φ′ ∈ TPL suchthat α T≤ T

∨(Φ′), Φ′ T⊆ T ↓L Tη(β) and Φ′ T⊆ T ↓L Φ. Define Γ := λT (Φ′),

then it follows from the definition of /L0 that β /0 Γ; moreover, it now follows

from Lemma 5.3.24 that Γ ⊆ ↓TLβ ∩ ↓TL λT (Φ). We conclude that (5.34) holds.Since α ∈ TL and Φ ∈ TPL were arbitrary, it follows that /L

0 is compatible withthe order T≤, so that 〈TL, T≤, /L

0 〉 is a flat site.

Having established that 〈TL, T≤, /L0 〉 is a flat site, we will now prove that it

presents VTL, i.e. that VTL ' Fr〈TL, T≤, /L0 〉.

5.3.26. Theorem. Let L be a frame and let T be a standard, finitary, weakpullback-preserving functor. Then VTL admits the following flat site presentation:

VTL ' Fr〈TL, T≤, /L0 〉,

where /L0 = (α, λT (Φ)) ∈ L× PTL | α T≤ T

∨(Φ), and in each direction, the

isomorphism is the unique frame homomorphism extending the identity map idTLon the set of generators of VTL and Fr〈TL, T≤, /L

0 〉, respectively.

Proof For this proof, we denote the insertion of generators from TL to VTL by∇, and from TL to Fr〈TL, T≤, /L

0 〉 by ♥. We will show that

1. the function ♥ : TL→ Fr〈TL, T≤, /L0 〉 is compatible with the relations (∇1),

(∇2′) and (∇3), and

2. that the function ∇ : TL→ VTL has the following properties:

(a) ∇ is order-preserving;

(b) 1 =∨∇α | α ∈ TL;

(c) for all α, β ∈ TL, ∇α ∧∇β =∧∇γ | δ T≤ α, β;

(d) for all α /L0 Γ, ∇α ≤

∨∇β | β ∈ Γ.

(1). First consider (∇1). Suppose that α, β ∈ TL such that α T≤ β; wehave to show that ♥α ≤ ♥β. This follows immediately from the fact that♥ : TL → Fr〈TL, T≤, /L

0 〉 is order-preserving. Secondly, consider (∇2′). LetΓ ∈ PωTL, we then have to show that∧

γ∈Γ♥γ ≤∨♥δ | ∀γ ∈ Γ, δ T≤ γ. (5.35)


Recall from §5.2.4 that since 〈TL, T≤, /L0 〉 is a flat site, we know that 1 =

∨♥α |

α ∈ TL and that for all α, β ∈ TL, ♥α ∧ ♥β =∧♥γ | δ T≤ α, β. It now

follows by induction on the size of Γ that (5.35) holds.Finally for (∇3), take Φ ∈ TPL. We have to show that ♥T

∨(Φ) ≤

∨♥β |

β ∈ λT (Φ). This follows immediately from the definition of /L0 , since T

∨(Φ) T≤

T∨

(Φ). We conclude that ♥ : TL→ Fr〈TL, T≤, /L0 〉 is compatible with the rela-

tions (∇1), (∇2) and (∇3) and thus there must be a unique frame homomorphismf : VTL→ Fr〈TL, T≤, /L

0 〉 which extends ♥.(2). We first have to show that ∇ is order-preserving, i.e. that if α T≤ β,

then ∇α ≤ ∇β. This follows immediately from (∇1). Secondly, we must showthat (2)(b) and (2)(c) are satisfied, but this follows immediately from (∇2′).Finally, consider (2)(d), i.e. suppose that α /L

0 Γ. By definition of /L0 , there is some

Φ ∈ TPL such that α T≤ T∨

(Φ) and λT (Φ) = Γ. Now we need to show that∇α ≤

∨∇β | β ∈ λT (Φ). This is easy to see, since

∇α ≤ ∇T∨

(Φ) by (∇1),

≤∨∇β | β ∈ λT (Φ) by (∇3).

It follows that (2)(d) holds; consequently, there is a unique frame homomorphismg : Fr〈TL, T≤, /L

0 〉 → VTL extending ∇.Finally, it is easy to see that

gf = idVTL and fg = id〈TL,T≤,/L0〉,

so that indeed VTL ' Fr〈TL, T≤, /L0 〉.

In light of Theorem 5.3.26 above, we denote the insertion of generators by∇ : TL → Fr〈TL, T≤, /L

0 〉. We now arrive at the most important corollary ofTheorem 5.3.26, which says that every element of VTL has a disjunctive normalform.

5.3.27. Corollary. Let L be a frame. Then for all x ∈ VTL, there is a Γ ∈ PTLsuch that x =

∨∇γ | γ ∈ Γ.

Proof By Theorem 5.3.26 we know that VTL ' SupLat〈TL, T≤, /L0 〉. The state-

ment now follows Fact 5.2.14.

5.3.28. Remark. It is not hard to show that

SupLat〈TL, T≤, /L0 〉 ' SupLat〈TL | (∇1), (∇3)〉.

Consequently, by Theorem 5.3.26 and Fact 5.2.14, the order on VTL is uniquelydetermined by the relations (∇1) and (∇3).

5.4. Preservation results 185

5.4 Preservation results

Now that we have established the T -powerlocale construction, we can set aboutto prove that it is well-behaved. One particular kind of good behaviour is to askthat it preserves algebraic properties. In this section, we present several initialresults in this area. In particular, we show that VT preserves regularity and zero-dimensionality of frames, and the property of being a compact zero-dimensionalframe.

5.4.1 Regularity and zero-dimensionality

The purpose of this subsection is to prove that the operation VT preserves regularityand zero-dimensionality of frames. Both of these notions are defined in terms ofthe well-inside relation 0; accordingly, the main technical result of this subsectionstates that if α T0 β, then also ∇α 0VTL ∇β. We first recall some notions leadingup to the definition of regularity.

5.4.1. Definition. Given two elements a, b of a distributive lattice L, we saythat a is well inside b, notation: a 0 b, if there is some c in L such that a ∧ c = 0and b ∨ c = 1. If a 0 a we say a is clopen. We denote the clopen elements of L byCL.

In case L is a frame, in the definition of 0, for the element c witnessing thata 0 b we may always take the Heyting complementation ¬a of a. In other words,a 0 b iff b∨¬a = 1. Consequently, if a is clopen then a∨¬a = 1. In the sequel wewill use not only this fact, but also the following properties of 0 without warning.

5.4.2. Fact ([54], L. III.1.1). Let L be a frame.

1. 0 ⊆ ≤;

2. ≤ ;0 ;≤ ⊆ 0;

3. for X ∈ PL, if ∀x ∈ X.x 0 y then∨X 0 y;

4. for X ∈ PL, if ∀x ∈ X.y 0 x then y 0∧X;

5. a 0 a iff a has a complement.

5.4.3. Definition. A frame L is regular if every a ∈ L satisfies

a =∨b ∈ L | b 0 a.

We say L is zero-dimensional if for all a ∈ L,

a =∨b ∈ CL | b ≤ a.


We record the following useful property of CL [54, §III-1.1]:

5.4.4. Fact. Let L be a frame. Then 〈CL,∧,∨, 0, 1〉 is a sublattice of L.

We define a function ⇓ : PL→ PCL which maps A ∈ PL to ↓A ∩ CL.

5.4.5. Lemma. If L is a zero-dimensional frame, then

1. ∀α ∈ TL, ∇α =∨∇β | β ∈ TCL, β T≤ α;

2. ∀Φ ∈ TPL, T∨

(Φ) = T∨ T ⇓(Φ);

3. ∀Φ ∈ TPL, ∀α ∈ TL, [α ∈ TCL and α T≤ ; T∈ Φ] iff α ∈ λT (T ⇓(Φ)).

Similarly to (1), if L is regular then ∀α ∈ TL, ∇α =∨∇β | β ∈ TL, β T0 α.

Proof (1). First, observe that for all a ∈ L, we have that

a =∨b ∈ CL | b ≤ a by zero-dimensionality,

=∨⇓a by definition of ⇓,

=∨⇓ η(a) by def. of η : IdSet → P .

By relation lifting, it follows that

∀α ∈ TL, α = T∨ T ⇓ Tη(α). (5.36)

Now observe that for all a, b ∈ L, we have b ∈ ⇓a iff b ∈ CL and b ≤ a. Byrelation lifting, it follows that

∀α, β ∈ TL,[β T∈ T ⇓ Tη(α) iff β ∈ TCL and β T≤ α

]. (5.37)

Combining these two observations, we see that

∇α = ∇ (T∨ T ⇓ Tη(α)) by (5.36),

=∨∇β | β T∈ T ⇓ Tη(α) by (∇3),

=∨∇β | β ∈ TCL, β T≤ α by (5.37).

(2). It follows by zero-dimensionality of L that for all A ∈ PL, we have∨A =

∨⇓A. Consequently, by relation lifting, it follows that (2) holds.

(3). Take a ∈ L and A ∈ PL. Then

a ∈ ⇓A⇔ a ∈ CL and ∃b ∈ A, a ≤ b by definition of ⇓,

⇔ a ∈ CL and a ≤ ; ∈ A by def. of relation composition.

It follows by relation lifting that

∀Φ ∈ TPL, ∀α ∈ TL, α T∈ T ⇓(Φ) iff α ∈ TCL and α T≤ ; T∈ Φ.


Now it follows by definition of λT (Φ) that (3) holds.For the last part of the proof, first observe that if L is regular, then for all

a ∈ L, a =∨w(a), where we temporarily define w : L→ PL as

w : a 7→ b ∈ L | b 0 a.


T∨ Tw = idL . (5.38)

Moreover, it follows by definition of w : L→ PL that for all a, b ∈ L, b ∈ w(a) iffb 0 a. Consequently,

∀α, β ∈ TL, β T∈ Tw(α) iff β T0 α. (5.39)

Now we see that for any α ∈ TL,

∇α = ∇ (T∨ Tw(α)) by (5.38),

=∨∇β | β T∈ Tw(α) by (∇3),

=∨∇β | β T0 α by (5.39).

The key technical lemma of this subsection states that relation lifting preservesthe 0-relation.

5.4.6. Lemma. Let T be a standard, finitary, weak pullback-preserving functorand let L be a frame. Then

for all α, β ∈ TL : α T0 β implies ∇α 0VTL ∇β. (5.40)

Proof Let α, β ∈ TL be such that α T0 β. Our aim will be to show that∇α 0VTL ∇β.

We may assume without loss of generality that

∃f : Base(α)→ Base(β) such that β = (Tf)α and ∀a ∈ Base(α), a 0 fa.(5.41)

To justify this assumption, assume that we have a proof of (5.40) for all βsatisfying (5.41). To derive (5.40) in the general case, consider arbitrary elementsα, β′ ∈ TL such that α T0 β′. In order to show that ∇α T0 ∇β′, consider themap f : Base(α) → L given by f(a) :=

∧b ∈ Base(β′) | a 0 b. On the basis

of Fact 5.4.2 it is not difficult to see that Gr(f) ⊆ 0 and so by the properties ofrelation lifting we obtain Gr(Tf) ⊆ T0. In particular, we find that α T0 (Tf)α;thus by our assumption we may conclude that ∇α 0 ∇(Tf)α. Also, observethat a 0 b implies fa ≤ b, for all a ∈ Base(α) and b ∈ Base(β′). Hence byLemma 5.2.8 we may conclude from α T0 β′ that (Tf)α T≤ β′, which gives∇(Tf)α ≤ ∇β′. Combining our observations thus far, by Fact 5.4.2 it follows from


∇α 0 ∇(Tf)α and ∇(Tf)α ≤ ∇β′ that ∇α 0 ∇β′ indeed. Thus our assumption(5.41) is justified indeed.

Turning to the proof itself, consider the map h : PBase(α)→ L given by

h(A) :=∧

(¬a | a ∈ A ∪ fa | a 6∈ A) .

Our first observation is that, since by assumption ¬a ∨ fa = 1L for each a ∈Base(α), we may infer that

1L =∧¬a ∨ fa | a ∈ Base(α),

a straightforward application of the (finitary) distributive law yields that

1L =∨h(A) | A ∈ PBase(α). (5.42)

Define X ⊆ L to be the range of h, so that we may think of h as a surjection h :PBase(α)→ X, and read (5.42) as saying that 1 =

∨X. Using Lemma 5.3.10(5),

from the latter observation we may infer that

1VTL =∨∇ξ | ξ ∈ TX. (5.43)

However, from h : PBase(α) → X being surjective we may infer that Th :TPBase(α)→ TX is also surjective, so that we may read (5.43) as

1VTL =∨∇Th(Φ) | Φ ∈ TPBase(α). (5.44)

This leads us to the key observation in our proof: We may partition the setTh(Φ) | Φ ∈ TPBase(α) into elements γ such that ∇γ ≤ ∇β, and elements γsatisfying ∇α ∧∇γ = 0VTL.

1. Claim. Let Φ ∈ TPBase(α).(a) If (α,Φ) ∈ T 6∈, then Th(Φ) T≤ β;(b) if (α,Φ) 6∈ T 6∈, then ∇α ∧∇Th(Φ) = 0VTL.

Proof of Claim For part (a), it is not hard to see that

a 6∈ A⇒ h(A) ≤ f(a), for all a ∈ Base(α), A ∈ PBase(α).

From this it follows by Lemma 5.2.8 that

α T 6∈ Φ⇒ Th(Φ) T≤ (Tf)(α) = β.

For part (b), assume that ∇α∧∇Th(Φ) > 0VTL. It suffices to derive from thisthat α T 6∈ Φ.


Let ≤′ be the restriction of ≤ to the non-zero part of L, that is, ≤′ := ≤L′×L′ ,where L′ = L \ 0L. We claim that for all γ, δ ∈ TL:

∇γ ∧∇δ > 0VTL ⇒ (γ, δ) ∈ T≥′ ; T≤′. (5.45)

To see this, assume that ∇γ ∧∇δ > 0VTL, and observe that Lemma 5.3.5 yieldsthe existence of a θ ∈ TL such that ∇θ > 0VTL and θ T≤ γ, δ. It follows fromLemma 5.3.10(1) that γ, δ and θ all belong to TL′, and so θ is witnesses to thefact that (γ, δ) ∈ T≥′ ; T≤′.

By (5.45) and the assumption on α and Φ it follows that (α,Φ) ∈ T≥′ ; T≤′ ;(GrTh) , and so by Fact 5.2.7 we obtain

(α,Φ) ∈ T (≥′ ;≤′ ; (Grh) ) (5.46)

The crucial observation now is that

≥′ ;≤′ ; (Grh) ⊆ 6∈. (5.47)

For a proof, take a pair (a,A) ∈ L× PL in the LHS of (5.47), and suppose forcontradiction that a ∈ A. Then by definition of h we obtain h(A) ≤ ¬a, so thata ∧ h(A) = 0L. But if a ≥′ ;≤′ ; (Grh) A, then there must be some b such thatb ≤′ a, h(A), and by definition of ≤′ this can only be the case if b > 0L. This givesthe desired contradiction.

Finally, by monotonicity of relation lifting, it is an immediate consequence of(5.46) and (5.47) that α T 6∈ Φ. This finishes the proof of the Claim.

On the basis of the Claim it is straightforward to finish the proof. Define

c :=∨

Th(Φ) | Φ ∈ TPBase(α) such that (α,Φ) 6∈ T 6∈,

then we may calculate that

c ∨∇β

≥ c ∨∨

Th(Φ) | Φ ∈ TPBase(α) such that (α,Φ) ∈ T 6∈

(Claim 1(a))

=∨Th(Φ) | Φ ∈ TPBase(α) (definition of c)

= 1VTL (equation (5.44))

and

∇α ∧ c

=∨∇α ∧ Th(Φ) | Φ ∈ TPBase(α) such that (α,Φ) 6∈ T 6∈

(distributivity)

=∨

0VTL | Φ ∈ TPBase(α) such that (α,Φ) 6∈ T 6∈

(Claim 1(b))

= 0VTL

In other words, c witnesses that ∇α 0VTL ∇β.


We now arrive at the main result of this subsection, namely, that the T -powerlocale construction preserves regularity and zero-dimensionality.

5.4.7. Theorem. Let L be a frame and let T be a standard, finitary, weakpullback-preserving functor.

1. If L is regular then so is VTL.

2. If L is zero-dimensional then so is VTL.

Proof (1). By Corollary 5.3.27, it suffices to show that for all α ∈ TL,

∇α =∨∇β ∈ VTL | ∇β 0 ∇α. (5.48)

Take α ∈ TL; we see that

∇α =∨∇β | β T0 α by Lemma 5.4.5,

≤∨∇β | ∇β 0VTL ∇α by Lemma 5.4.6,

≤ ∇α since 0 ⊆ ≤.

It follows that (5.48) holds, concluding the proof of part (1).

(2). Again by Corollary 5.3.27, it suffices to show that for all α ∈ TL,

∇α =∨∇β | ∇β ∈ CVTL, ∇β ≤ ∇α. (5.49)

The main observation here is that

∀β ∈ TCL, ∇β ∈ CVTL. (5.50)

To see why, recall that CL := b ∈ L | b 0 b, so that for all b ∈ CL, b = b impliesb 0 b. Consequently, by relation lifting,

∀β ∈ TCL, β T0 β.

It follows by Lemma 5.4.6 that (5.50) holds. Now

∇α =∨∇β | β ∈ TCL, β T≤ α by Lemma 5.4.5(1),

≤∨∇β ∈ CVTL | β T≤ α by (5.50),

≤∨∇β ∈ CVTL | ∇β ≤ ∇α by (∇1),

= ∇α by order theory.

It now follows that (5.49) holds; consequently we see that (2) holds.


5.4.2 Compactness

In this subsection, we will show that if L is compact and zero-dimensional, then sois VTL. Our proof strategy is as follows. Given a compact zero-dimensional frameL, we will define a new construction V C

T L which is guaranteed to be compact, andthen we show that VTL ' V C

T L.We define a flat site presentation 〈TCL, T≤, /C0 〉, where

/C0 := (α, λT (Φ)) ∈ TCL × PTL | α T≤ T∨

(Φ), Φ ∈ TPωCL.

Observe that we view TCL as a substructure of TL, which is justified by thefact that CL is a sublattice of L (Fact 5.4.4): this fact tells us that

∨: PL→ L

restricts to a function from PωCL to CL; consequently, by standardness of T T∨

maps TPωCL to TCL. Below, we will need the following property of relation liftingwith respect to ordered sets.

5.4.8. Lemma. Let P be a poset with a top element 1.

∀β ∈ TP, ∃α ∈ T1, β T≤ α;

Proof Consider the following function at the ground level: f : P → 1, wheref is the constant function f : b 7→ 1. Then for all b ∈ P , we have b ≤ f(b) andf(b) ∈ 1. By relation lifting, we see that for all β ∈ TP , β T≤ Tf(β) andTf(β) ∈ T1. The statement follows.

5.4.9. Lemma. Let L be a frame. Then 〈TCL, T≤, /C0 〉 is a flat site. Moreover,if T maps finite sets to finite sets then Fr〈TCL, T≤, /C0 〉 is a compact frame.

Proof Because CL is a meet-subsemilattice of L, we can apply Lemma 5.3.23to TCL. Now the proof that 〈TCL, T≤, /C0 〉 is a flat site is analogous to that ofLemma 5.3.25.

Now suppose that T maps finite sets to finite sets. Then for all Φ ∈ TPωCL, itfollows by Fact 5.2.12(3) that λT (Φ) is finite. Consequently,

∀α /C0 λT (Φ), λT (Φ) is finite.

Moreover, by Lemma 5.4.8,

TCL = ↓TCLT1L,

since 1L ∈ CL as CL is a sublattice of L. Now since we assumed that T maps finitesets to finite sets, 1L must be finite. It is now follows from a straight-forwardgeneralization of [93, Proposition 11] that Fr〈TCL, T≤, /C0 〉 is a compact frame.(The only change we need to make to [93, Proposition 11] is to generalize fromfrom using single finite trees to using disjoint unions of |T1L|-many trees, sothat one can cover each element of T1L.)


We define V CT L := Fr〈TCL, T≤, /C0 〉, and for the time being we denote the

insertion of generaters by ♥ : TCL → V CT L. Our goal is now to show that VTL '

V CT L. We will use a shortcut, exploiting the fact that both VTL and V C

T L haveflatsite presentations: we will define suplattice homomorphisms f ′ : VTL→ V C

T Land g′ : V C

T L→ VTL. We then show that g′ f ′ = id and f ′ g′ = id, so that VTLand V C

T L are isomorphic as suplattices. It then follows from order theory that theyare also isomorphic as frames. We start by defining a function g : TCL → VTL,defined as

g : α 7→ ∇α.

5.4.10. Lemma. Let L be a frame. Then the fuction g defined above extends to asuplattice homomorphism g′ : V C

T L→ VTL such that g′ ♥ = g.

V CT L

g′ // VTL

TCL

♥OO

g

<<xxxxxxxxx

Proof We need to show that g : TCL → VTL preserves the order on TCL andpreserves covers in to joins: if α /C0 λT (Φ), where α ∈ TCL, Φ ∈ TPCL andα T≤

∨(Φ), then g(α) ≤

∨g(β) | β ∈ λT (Φ). Both of these properties

follow straightforwardly from the fact that 〈TCL, T≤, /C0 〉 is a substructure of〈TL, T≤, /L

0 〉.

The next step is to define the suplattice homomorphism f ′ : VTL→ V CT L. This

requires a little more work then the definition of g′ : V CT L→ VTL, beginning with

the following lemma.

5.4.11. Lemma. Let L be a compact frame. If α ∈ TCL and Φ ∈ TPCL suchthat α T≤ T

∨(Φ), then there exists Φα ∈ TPωCL such that Φα T⊆L Φ and

α T≤ T∨

(Φα).

Proof Since L is compact, we can show that

for all a ∈ CL, a is compact. (5.51)

After all, if a ∈ CL and A ∈ PL such that a ≤∨A, then also 1 ≤ a ∨ ¬a ≤∨

A ∪ ¬a, so by compactness of L, there exists a finite A′ ⊆ A such thata ∨ ¬a ≤

∨A′ ∪ ¬a. Consequently, a ≤

∨A′. Since A was arbitrary, it follows

that a is compact.We define

S := (≤ ;Gr(∨

) ) CL×PCL ;

so that (a,A) ∈ S iff a ∈ CL, A ∈ PCL and a ≤∨A. By (5.51), we can define a

function h : S → S where h : (a,A) 7→ (a′, A′) such that a = a′, A′ ⊆ A, a′ ≤∨A′


(otherwise h would not be well-defined) and such that A′ is finite, i.e. A′ ∈ PωCL.In other words, h : S → S is a function which assigns a finite subcover A′ to a setof zero-dimensional opens A covering a zero-dimensional open element a . If wedenote the projection functions of S as

CL Sp1oo p2 // PCL

then we can encode the above-mentioned properties of h as follows:

∀x ∈ S, p1 h(x) = p1(x);

∀x ∈ S, p2 h(x) ⊆ p2(x);

∀x ∈ S, p2 h(x) ∈ PωCL.


∀x ∈ TS, Tp1 Th(x) = Tp1(x); (5.52)

∀x ∈ TS, Tp2 Th(x) T⊆ Tp2(x); (5.53)

∀x ∈ TS, Tp2 Th(x) ∈ TPωCL. (5.54)

Finally, observe that it follows by relation lifing that

∀α ∈ TCL,∀Φ ∈ TPCL, α T≤∨

(Φ) iff α TS Φ.

Now take α ∈ TCL and Φ ∈ TPCL such that α T≤∨

(Φ). Then by the above,we have α TS Φ, so by definition of T there must exist some x ∈ TS suchthat Tp1(x) = α and Tp2(x) = Φ. We define Φα := Tp2 Th(x); observe thatTp1 Th(x) = Tp1(x) = α by (5.52). Since Th is a function from TS to TS, wesee that α TS Φα, so that α T≤ T

∨(Φα). Moreover by (5.53) Φα T⊆ Φ and by

(5.54), Φα ∈ TPωCL. This concludes the proof.

We now define a map f : TL→ V CT L by sending

f : α 7→∨♥β | β ∈ TCL, β T≤ α.

This will give us our suplattice homomorphism f ′ : VTL→ V CT L.

5.4.12. Lemma. If L is a compact zero-dimensional frame then f : TL→ V CT L

defined above extends to a suplattice homomorphism f ′ : VTL → V CT L, where

f ′ ∇ = f .

VTLf ′ // V C

T L

TL

∇

OO

f

;;wwwwwwww


Proof In order to show that f : TL → V CT L extends to a suplattice homomor-

phism, we need to show that f preserves the order on TL and f transformscovers into joins, i.e. that for all (α, λT (Φ)) ∈ /0, where α T≤ T

∨(Φ), we have

f(α) ≤∨f(γ) | γ ∈ λT (Φ). To see why f is order-preserving, suppose that

α0, α1 ∈ TL and that α0 T≤ α1. Then

f(α0) =∨♥β | β ∈ TCL, β T≤ α0 by definition of f ,

≤∨♥β | β ∈ TCL, β T≤ α1 since β T≤ α0 T≤ α1 ⇒ β T≤ α1,

= f(α1) by definition of f .

Before we go ahead and show that f tranforms covers α /0 λT (Φ) into joins, we

show that the expression∨f(γ) | γ ∈ λT (Φ) can be simplified:

∀Φ ∈ TPL,∨f(γ) | γ ∈ λT (Φ) =

∨♥β | β ∈ λT (T ⇓(Φ)). (5.55)

To see why, observe that∨f(γ) | γ ∈ λT (Φ)

=∨∨

♥β | β ∈ TCL, β ≤ γ | γ ∈ λT (Φ)

by definition of f ,

=∨∨

♥β | β ∈ TCL, β ≤ γ | γ T∈ Φ

by definition of λT ,

=∨♥β | β ∈ TCL, ∃γ T∈ Φ, β ≤ γ by associativity of

∨,

=∨♥β | β ∈ TCL, β T≤ ; T∈ Φ by def. of relation composition,

=∨♥β | β ∈ λT (T ⇓(Φ)) by Lemma 5.4.5(3).

Let α ∈ TL and Φ ∈ TPL such that α T≤ T∨

(Φ); we need to show thatf(α) ≤

∨f(γ) | γ ∈ λT (Φ). By (5.55) it suffices to show that

f(α) ≤∨♥γ | γ ∈ λT (T ⇓(Φ)). (5.56)

Recall that f(α) =∨♥β | β ∈ TCL, β ≤ α. We will show that

∀β ∈ TCL, β T≤ α⇒ ♥β ≤∨♥γ | γ ∈ λT (T ⇓(Φ)). (5.57)

Suppose that β ∈ TCL and that β T≤ α. Then since we assumed that α T≤T∨

(Φ), it follows that β T≤ T∨

(Φ). By Lemma 5.4.5(2), we know that T∨

(Φ) =T∨ T ⇓(Φ), so we see that

β T≤ T∨ T ⇓(Φ).

Now since T ⇓(Φ) ∈ TPCL, we can now apply Lemma 5.4.11 to conclude thatthere must be some Φ′ ∈ TPωCL such that Φ′ T⊆ T ⇓(Φ) and β T≤

∨Φ′. Now it

follows by definition of /C0 that β /C0 λT (Φ′). Now

♥β ≤∨♥γ | γ ∈ λT (Φ′) since β /C0 λ

T (Φ′),

≤∨♥γ | γ ∈ λT (T ⇓(Φ)) by L. 5.3.24 since Φ′ T⊆ T ⇓(Φ).


Since β ∈ TCL was arbitrary it follows that (5.57) holds; consequently, (5.56) holdsso that we may indeed conclude that f transforms covers into joins. We concludethat f : TL→ V C

T L extends to a suplattice homomorphism f ′ : VTL→ V CT L.

Now that we have established the existence of suplattice homomorphismsf ′ : VTL→ V C

T L and g′ : V CT L→ VTL, we are ready to prove the theorem of this

subsection.

5.4.13. Theorem. Let T : Set → Set be a standard, finitary, weak pullback-preserving set funtor which maps finite sets to finite sets and let L be a frame. IfL is compact and zero-dimensional then so is VTL.

Proof It follows by Theorem 5.4.7 that VTL is zero-dimensional. To show thatVTL is compact, it suffices to show that VTL ' V C

T L by Lemma 5.4.9. We willestablish that VTL ' V C

T L by showing that g′ : V CT L→ VTL and f ′ : VTL→ V C

T Lare suplattice isomorphisms, because g′ f ′ = idVTL and f ′ g′ = idV CT L. Thisis sufficient since by order theory, any suplattice isomorphism is also a frameisomorphism. We begin by making the following claim:

∀α ∈ TL, g′ f(α) = ∇α. (5.58)

After all, if α ∈ TL then

g′ f(α) = g′(∨♥β | β ∈ TCL, β T≤ α

)by definition of f ,

=∨g′(♥β) | β ∈ TCL, β T≤ α since g′ preserves

∨,

=∨g(β) | β ∈ TCL, β T≤ α by Lemma 5.4.10,

=∨∇β | β ∈ TCL, β T≤ α by definition of g,

= ∇α by Lemma 5.4.5(1).

It follows that (5.58) holds. Conversely, we claim that

∀α ∈ TCL, f′ g(α) = ♥α. (5.59)

This is also not hard to see. Take α ∈ TCL, then

f ′ g(α) = f ′ (∇α) by definition of g,

= f(α) by Lemma 5.4.12,

=∨♥β | β ∈ TCL, β T≤ α by definition of f ,

= ♥α since ♥ is order-preserving.

It follows that (5.59) holds. Now we see that for all α ∈ TL,

g′ f ′(∇α) = g′ f(α) since f ′ ∇ = f ,

= ∇α by (5.58),

= idVTL (∇α) .


In other words, we see that g′ f ′ and idVTL agree on the generators of VTL; itfollows that g′ f ′ = idVTL. An analogous argument shows that f ′ g′ = idV CT L.

We conclude that VTL and V CT L are isomorphic as suplattices and consequently

also as frames; it follows that VTL is compact.

5.5 Further work

In this final section of this chapter, we give a list of possible questions for furtherwork. The first question, or rather, task, is to investigate concrete instances of theT -powerlocale, beyond the cases T = Pω and T = Id. Some other, more technicalquestions could be the following:

• A straightforward question is if we can prove, given reasonable assumptionsabout T , that VT preserves compactness.

• For a follow-up question, consider the following. There is a dual equivalencebetween KRegFrm, the category of compact regular frames and framehomomorphisms, and KHaus, the category of compact Hausdorff spacesand continuous maps, as witnessed by the functors pt : KRegFr→ KHausand Ω: KHaus→ KRegFr. If we denote the classical Vietoris hyperspacefunctor by K : KHaus → KHaus, then it is the case that K pt 'pt V . (All of the above can be found in [54, Ch. III].) Our question isnow if it is possible, given a coalgebra functor T : Set → Set such thatVT preserves compactness (and regularity, by Theorem 5.4.7), to define afunctor KT : KHaus→ KHaus, such that KT pt ' pt VT . Palmigiano& Venema propose an approach to this problem via Chu spaces in [73].

• A related question concerns the work in [66, 67]. In these papers, the authorsdefine a functor MT : BA→ BA on Boolean algebras, similar to the way wedefined VT : Fr→ Fr on frames. It is known [54, §§II-3 and II-4] that L is acompact and zero-dimensional frame iff L ' I B for some Boolean algebra B,where I is the ideal completion functor. This raises the question if our VTis equivalent to MT of [67] modulo ideal completion, i.e. if I MT ' VT I.

• A rather technical question is the one raised in Remark 5.3.22: in whatsense, if at all, is ρ : VT → VT ′ a right adjoint?

A different line of questions concerns constructive mathematics. In their currentform, it is not entirely obvious where in the results we have presented there mightbe constructive issues. We mention two possible improvements:

• We have described the construction VTL by taking a frame L and then givinga presentation of VTL. One could also define VT on presentations of frames,rather than on the full frames being presented. This way, one can study

5.5. Further work 197

frames entirely via their presentations, which is conceptually comparable tothe approach of formal topology [25].

• Our results crucially use relation lifting, which seems to be tied to thecategory of sets rather much. If one would take a more axiomatic approachto the properties of λT : TP → PT , it might be possible to escape Set anddescribe yet another generalization of VT which can be constructed usingother categories.

Appendix A

Preliminaries

In this appendix, we will briefly discuss some of the mathematical backgroundknowledge that we rely on elsewhere in this dissertation. The presentation of thisappendix is not linear: when explaining one subject, we will sometimes refer toanother one which may lie further ahead in the text.

A.1 Set theory

Throughout this dissertation, we assume that the reader is familiar with elementaryset theoretical notions such as membership x ∈ X, intersection X ∩ Y , unionX ∪ Y and set theoretic difference X \ Y . An exception is Chapter 5, where weuse some extra notation which does not occur in the rest of the dissertation. Theadditional preliminaries for Chapter 5 are discussed in §5.2.

The only thing we would like to briefly mention at this point is the power-set construction. If f : X → Y is a function between sets X and Y , then byf−1 : P(Y )→ P(X) we denote the inverse image function, which maps any setU ⊆ Y to

f−1(U) := x ∈ X | f(x) ∈ U.

At the same time, if U ⊆ X, then we define

f [U ] := f(x) | x ∈ U;

this yields a function f [·] : P(X)→ P(Y ). Using these two mappings on subsetsbased on a function f : X → Y , we can define two functors on Set, the categoryof sets, with the same action on objects, viz. sending X to P(X). The covariantpowerset functor P : Set→ Set maps f : X → Y to Pf : P(X)→ P(Y ), wherePf : U 7→ f [U ]. The contravariant powerset functor P : Set → Setop mapsf : X → Y to P f : P(Y )→ P(X), where P f : U 7→ f−1(U).

199

200 Appendix A. Preliminaries

A.2 Category theory

Our main reference for category theory is Mac Lane [69]; alternatively, one canconsult Adamek, Herrlich & Strecker [3].

A.2.1 Categories and functors

A category is a structure C consisting of a class of objects X, Y, Z, . . . and a binaryfunction HomC which assigns to any two objects X, Y a class of morphisms, orarrows, denoted HomC(X, Y ).1 If f ∈ HomC(X, Y ) then we write f : X → Y or

Xf−→ Y.

We require that if (X, Y ) 6= (X ′, Y ′), then HomC(X, Y ) and HomC(X ′, Y ′) aredisjoint. In other words, given an arrow f in C, f has a unique domain X andcodomain Y such that f ∈ HomC(X, Y ). Every category comes equipped withan associative composition operation for morphisms, and identity morphismsidX , one for each object X. If f : X → Y and g : Y → Z then g f : X → Z is amorphism from X to Z. Saying that composition is associative means that for allf : X → Y , g : Y → Z and h : Z → W , we have

h (g f) = (h g) f.

The identity arrows idX are characterized by the following property: for all arrowsf : X → Y and g : Y → X,

f idX = f and idX g = g.

We also use the arrows to define the notion of isomorphism. An arrow f : X → Yis an isomorphism if there exists an arrow g : Y → X such that

gf = idX and fg = idY .

A.2.1. Example. As a prototypical example of a category, consider Set, whichhas as its objects the class of all sets, and as its arrows all functions between sets.Composition of arrows is then simply composition of functions; the identity arrowsare the identity functions and isomorphisms are precisely the bijective functions.Note that this is not the prototypical example of a category, see [69, §I.2] for alist of further basic examples.

Let C and C′ be categories. We say C′ is a subcategory of C if every objectof C′ is also an object of C, and if for all objects X, Y of C′, HomC′(X, Y ) ⊆HomC(X, Y ). We say C′ is a full subcategory of C if for all objects X, Y of C′,HomC′(X, Y ) = HomC(X, Y ).

1In all categories we consider in this dissertation, HomC(X, Y ) is a set rather than a class.

A.2. Category theory 201

A.2.2. Example. The category Setf , which has as its objects the class of allfinite sets, and as its morphisms all functions between finite sets, forms a fullsubcategory of Set.

Given a category C, we can construct its dual category Cop. The objects ofCop are simply those of C. The arrows of Cop are in 1-1 correspondence f 7→ f op

with those of C. The only difference between C and Cop is that if f : X → Y isan arrow of C, then f op : Y → X in Cop goes in the other direction. We can nowdefine the composite of two arrows f op : Y → X and gop : Z → Y in Cop to be

f op gop := (gf)op.

It is easy to see that (idX)op is the identity arrow of X in Cop, and that Cop indeedforms a category.

Because of the way Cop is defined in terms of C, we can unravel statementsabout Cop into statements about C. Using this fact, we can automatically definethe dual version of any categorical concept. Usually, we will name such a dualconcept by prefixing ‘co-’ to the name of the original concept. This process canbe made much more precise, see [69, §II.1, II.2].

A.2.3. Example. Let C be a category and let F : C → C be an endofunctor,i.e. a functor with the same domain and codomain. An F -algebra consists of anobject X and a morphism h : F (X)→ X. The dual notion, that of an F -coalgebra,consists simply of an object X and a morphism r : X → F (X).

A functor is a structure-preserving map between categories. Concretely, ifC,D are categories then a functor F : C → D consists of an assignment of anobject F (X) to every object X of C, and of an assignment of an arrow F (f) ∈HomD(FX,FY ) to every arrow f ∈ HomC(X, Y ) such that F (gf) = F (g)F (f)

for all Xf−→ Y

g−→ Z in C and F (idX) = idF (X) for all X in C. If it is not visuallyconfusing, we will sometimes omit parentheses, writing e.g. Ff instead of F (f).

A.2.4. Example. As a trivial example of a functor, observe that given anycategory C we can define the identity functor IdC : C → C which leaves allobjects and arrows unchanged.

A functor F : C→ Dop is called a contravariant functor from C to D; one canalternatively view F as an ‘arrow-reversing’ functor from C to D. The notion offunctor we introduced before is sometimes also called a covariant functor.

A natural transformation ν between functors F,G : C → D is a family ofD-morphisms νX : F (X)→ G(X), one for each object X of C, such that for allf : X → Y in C, the following diagram commutes:

X

f

F (X)νX //

F (f)

G(X)

G(f)

Y F (Y ) νY

// G(Y )


If ν is a natural transformation such that for each X, νX is an isomorphism, thenwe call ν a natural isomorphism. If there exists a natural isomorphism betweentwo functors F,G : C→ D, we say F and G are naturally isomorphic.

A.2.2 Adjunctions of categories

One of the fundamental notions of category theory is that of adjunctions betweencategories. A pair of functors F : C→ D and G : D→ C is called an adjunction,abbreviated F a G, if there exist natural transformations η : IdC → GF andε : FG→ IdD such that for all f : X → GY , there exists a unique f ′ : F (X)→ Ysuch that f = Gf ′ ηX , and for all g : FX → Y , there exists a unique g′ : X →G(Y ) such that g = εY Fg′:

GF (X)

G(f ′)

F (X)

f ′

FG(Y )εY

""FFFFFFFFFG(Y )

X

ηX;;xxxxxxxxx

f// G(Y ) Y F (X)

F (g′)

OO

g// Y X

g′

OO

A dual adjunction between categories C and D is simply an adjunction betweenC and Dop.

We say to categories C,D are equivalent if there exist functors F : C → Dand G : D→ C such that GF is naturally isomorphic to IdC and FG is naturallyisomorphic to IdD. If both F and G are contravariant, we say C and D are duallyequivalent.

If D is a subcategory of C, then the inclusion of D into C forms a functorJ : D → C. If this functor has a left adjoint F : C → D, we say that D is areflective subcategory of C, and we call F : C→ D a reflector. Dually, if J : D→ Chas a right adjoint G : C→ D, we say that D is a co-reflective subcategory of C,and we call G : C→ D a co-reflector.

A.3 Order theory and domain theory

In this section we will discuss some of the order theory and domain theory thatwe employ in this dissertation. Our main reference for order theory is Davey &Priestley [29]; our main reference for domain theory is Abramsky & Jung [1].

A.3.1 Pre-orders and partial orders

A pre-order is a structure P = 〈P,≤〉 where ≤ ⊆ P ×P is a binary relation whichis

• reflexive: ∀x ∈ P, x ≤ x;

A.3. Order theory and domain theory 203

• transitive: ∀x, y, z ∈ P, if x ≤ y and y ≤ z, then x ≤ z.

(Note that we will sometimes write ‘x ∈ P’ rather than ‘x ∈ P ’.) Observe that apreorder 〈P,≤〉 can be seen as a small category with a set of objects P , such thatbetween any two objects x, y there is at most one arrow. From this perspective,we get that HomP(x, y) 6= ∅ iff x ≤ y. The notion of dual category now specializesto the notion of the order dual of a pre-order; we denote the order dual of P byPop. In other words, Pop := 〈P,≥〉, where x ≥ y :⇔ y ≤ x. It is not hard to seethat Pop is again a pre-order. If x ∈ P, we define ↓P x = y ∈ P | y ≤ x; ↑P xis defined dually. If it is clear what order we are dealing with we will omit thesubscripts on ↓ and ↑. For U ⊆ P , we define ↓U =

⋃x∈U ↓x. If U = ↓U we say

that U is a lower set ; dually, if U = ↑U then U is an upper set. An equivalentcharacterization of lower sets is to say that U is a lower set iff for all x ∈ U andall y ≤ x, we have y ∈ U .

A partial order (or partially ordered set, or poset) is a pre-order 〈P,≤〉 whichsatisfies the additional property that it is

• anti-symmetric: ∀x, y ∈ P, if x ≤ y and y ≤ x then x = y.

Although most of the order-theoretic notions we will discuss below also can bedefined for pre-orders (in fact, even for categories), we will restrict our attentionbelow to partially ordered sets.

Given two posets P and Q, a function f : P → Q is called order-preserving iffor all x, y ∈ P such that x ≤ y, we have f(x) ≤ f(y). (This is the order-theoreticspecialization of the categorical notion of being a functor.) We say f : P → Qis an order embedding if for all x, y ∈ P, x ≤ y iff f(x) ≤ f(y). One can showthat order embeddings between posets are necessarily injective maps. An orderisomorphism is a surjective order-embedding. By Pos we denote the category ofpartially ordered sets and order-preserving maps.

A.3.1. Fact. Let P,Q be posets and let f : P → Q be a function. The followingare equivalent:

1. f is order-preserving;

2. for all U ⊆ P , f [↓U ] ⊆ ↓ f [U ];

3. for all V ⊆ Q, if V is a lower set then so is f−1(V ).

Products of posets are defined point-wise: if Pi | i ∈ I is a collection ofposets, we define ∏

I

Pi =⟨∏

IPi,≤⟩,

where for x, y ∈∏

I Pi, we have x ≤ y iff for all i ∈ I, x(i) ≤i y(i).


A.3.2 Adjunctions of partially ordered sets

Since every poset is a category, we can specialize the categorical notion of adjunc-tion we introduced in §A.2.2 to posets.

A.3.2. Definition. Let f : P → Q and g : Q → P be order-preserving maps.We say f and g are an adjoint pair, abbreviated f a g, if one of the followingequivalent conditions holds:

• ∀x ∈ P , ∀y ∈ Q, f(x) ≤ y iff x ≤ g(y);

• idP ≤ g f and f g ≤ idQ.

Adjoint pairs have many attractive mathematical properties; we list a fewbelow. As a reference, consider [1, Prop. 3.1.12].

A.3.3. Fact. Suppose that f : P → Q and g : Q → P form an adjoint pair,i.e. f a g. Then

1. f preserves all existing suprema and g preserves all existing infima;

2. f is an order embedding iff g is surjective iff g f = idP;

3. f is surjective iff g is order embedding iff f g = idQ.

A.3.3 Dcpo’s

A non-empty subset U ⊆ P of a partially ordered set P is called directed if for allx, y ∈ U , there exists a z ∈ U such that x ≤ z and y ≤ z. A dcpo D = 〈D,≤, 〉 isa partial order such that for every directed non-empty U ⊆ D, U has a supremumor join in D, denoted

∨U . By Dcpo we denote the category of dcpo’s and Scott-

continuous functions, i.e. functions which preserve directed joins. As a generalreference for facts about dcpo’s, we suggest Abramsky & Jung [1]. We will seelater that Scott-continuity is indeed a continuity property. For now we record thefollowing fact about finite products of dcpo’s. As a reference for this fact, consider[1, Prop. 3.2.2 and Lemma 3.2.6]

A.3.4. Fact. Let D1, . . . ,Dn,E be dcpo’s. Then the poset-product D1 × · · · × Dnis also a dcpo. If f : D1× · · ·×Dn → E is a function, then f preserves all directedjoins in D1 × · · · × Dn iff f preserves directed joins in each coordinate.

A.4. Lattice theory 205

A.3.4 Order and topology

Given a partial order P, one can use the order to define several topologies on P , aswe do in §A.7. As a general reference, we suggest the Compendium of ContinuousLattices[45, Ch. 2 and 3]. We will mention a few properties of one such topology,the Scott topology.

A.3.5. Definition. Let P = 〈P,≤〉 be a poset. A set U ⊆ P is Scott-open iffP \ U is a lower set closed under all existing directed joins. By σ↑(P) we denotethe Scott topology of P [45, Ch. 2].

The following well-known fact states that Scott-continuity of maps betweendcpo’s is indeed a continuity property. As a reference, consider [1, Prop. 2.3.4].

A.3.6. Fact. A function f : D→ E between dcpo’s preserves directed joins iff itis continuous with respect to the Scott topology.

Many order-theoretic properties can be characterized topologically; this is amajor topic in this dissertation. As another example consider the following.

A.3.7. Example. Given a poset P = 〈P,≤〉 define the Alexandrov topology tobe the collection of all upper sets of P [54, §II-1.8]. A function f : P→ Q betweenposets is order-preserving iff f is continuous with respect to the Alexandrovtopology.

A.4 Lattice theory

A standard reference for lattice theory is Birkhoff [18]. Good introductory exposi-tions can also be found in e.g. [23], [29] and [54, Ch. I].

A.4.1 Semilattices and suplattices

A ∨-semilattice is an algebra (see §A.6) L = 〈L, 0,∨〉 with a binary operation ∨called ‘join’ and a constant 0 which satisfy the following equations:

(i) x ∨ x ≈ x (idempotence) (ii) x ∨ y ≈ y ∨ x (commutativity)(iii) x ∨ (y ∨ z) ≈ (x ∨ y) ∨ z (associativity)(iv) x ∨ 0 ≈ x (0 is the identity).

A ∨-semilattice is a poset under the ordering a ≤ b :⇔ a ∨ b = b. Alternatively,one can characterize ∨-semilattices as posets in which any finite subset has aleast upper bound; one can then define 0 to be the upper bound of the empty set,and a ∨ b to be the least upper bound of a, b. A ∨-semilattice homomorphismf : L→M is a map between semilattices L, M which preserves ∨ and 0, i.e. suchthat f(a ∨ b) = f(a) ∨ f(b) for all a, b ∈ L and such that f(0) = 0.


A complete ∨-semilattice or suplattice is a partially ordered set in whicheach subset has a least upper bound; alternatively, a suplattice is an algebraL = 〈L,

∨, 0〉, where

∨is an infinitary operation.

A.4.1. Fact. Let L and M be suplattices and let f : L→M be a function. Thefollowing are equivalent:

1. f preserves all non-empty joins;

2. f preserves binary joins and directed joins.

The order dual notion of a ∨-semilattice is that of a ∧-semilattice. In a∧-semilattice L = 〈L,∧, 1〉, the operation ∧ (meet) determines a partial ordera ≤ b :⇔ a ∧ b = a. Under this order a ∧ b is the greatest lower bound of a, b.

A.4.2 Lattices and complete lattices

A (bounded) lattice is an algebra L = 〈L,∧,∨, 0, 1〉 which is simultaneously a∧-semilattice and a ∨-semilattice. Alternatively, a lattice is a partially orderedset in which each finite subset has both a least upper bound and a greatest lowerbound.

A.4.2. Remark. Throughout this dissertation we assume that lattices are bounded,i.e. that a lattice L always has a least element 0L and a greatest element 1L, andthat lattice homomorphisms preserve 0L and 1L.

By Lat we denote the category of lattices and lattice homomorphisms; by CLatwe denote the category of complete lattices and complete homomorphisms. Thefollowing fact is the order-theoretic version of a well-known result from categorytheory, known as the Adjoint Functor Theorem. As a reference for this order-theoretic version, consider [1, Prop. 3.1.13].

A.4.3. Fact. If f : L→ M is a map between complete lattices which preservesall joins, then there exists a unique g : M→ L such that f and g form an adjointpair f a g.

A.4.3 Distributive lattices, Heyting algebras and Booleanalgebras

Introductory discussions of distributive lattices, Heyting algebras and Booleanalgebras can be found in Johnstone [54, Ch. I]. For additional technical detail,see e.g. Balbes & Dwinger [7] for distributive lattices and Heyting algebras andKoppelberg [63] for Boolean algebras.

A lattice L is distributive if it satisfies the equation x∧(y∨z) ≈ (x∧y)∨(x∨z),or equivalently, if it satisfies the equation x ∨ (y ∧ z) ≈ (x ∨ y) ∧ (x ∨ z). We

A.5. Completions 207

denote the category of distributive lattices and lattice homomorphisms by DL. IfL is complete, we say L satisfies the (∧,

∨)-distributive law (also known as the

infinite distributive law) if for all a ∈ L and for all S ⊆ L, a ∧∨S =

∨b∈S(a ∧ b).

Complete lattices which satisfy the (∧,∨

)-distributive law are known as frames,also see §5.2.4.

A Heyting algebra is an algebra A = 〈A,∧,∨,→, 0, 1〉 such that 〈A,∧,∨, 0, 1〉is a distributive lattice and→ is a binary operation, known as Heyting implication,such that

z ∧ x ≤ y iff z ≤ x→ y.

Alternatively, one can say that → has to satisfy the following axioms:(i) x→ x ≈ 1 (ii) x ∧ (x→ y) ≈ x ∧ y

(iii) y ∧ (x→ y) ≈ y (iv) x→ (y ∧ z) ≈ (x→ y) ∧ (x→ z)It is a fact of order-theory that adjoints of maps are unique; consequently, since→ is defined via an adjointness property with respect to ∧, a given distributivelattice L admits at most one Heyting implication such that 〈L,∧,∨,→, 0, 1〉 is aHeyting algebra. We denote the category of Heyting algebras and Heyting algebrahomomorphisms by HA.

A Boolean algebra is an algebra A = 〈A,∧,∨,¬, 0, 1〉 such that 〈A,∧,∨, 0, 1〉is a distributive lattice, and where ¬ is a unary operation, known as negation orcomplementation, satisfying the following equations:

x ∨ ¬x ≈ 1 and x ∧ ¬x ≈ 0.

As was the case with Heyting algebras, a distributive lattice L admits at mostone negation operation making it into a Boolean algebra. In fact, every Booleanalgebra is also a Heyting algebra, if we define

a→ b := ¬a ∨ b.We denote the category of Boolean algebras and Boolean algebra homomorphismsby BA.

A.5 Completions

In §A.3 and §A.4 we have seen several kinds of complete ordered structures.Completions of ordered structures, that is embeddings of ordered structures intocomplete structures play an important role in this dissertation. In this section wewill discuss some properties of the filter and ideal completions and the MacNeillecompletion.

A.5.1 The ideal (and filter) completion of a pre-order

The ideal completion is a construction that allows one to embed any partial orderinto a dcpo. As references we suggest Plotkin [75, §6] and Abramsky & Jung [1,§2.2.6].


Ideals and filters

A.5.1. Definition. Let P = 〈P,≤〉 be a partial order. An ideal of P is anon-empty, directed lower set I ⊆ P. We define I P = 〈Idl(P),⊆〉 (the idealcompletion of P) to be the collection of ideals of P ordered by subset inclusion.Dually, a filter is a non-empty upper set F ⊆ P which is co-directed. We defineF P := 〈Filt(P),⊇〉 (the filter completion of P) to be the collection of filters of P.

One can embed P in I P by mapping x 7→ ↓x. Moreover, given a order-preserving function f : P→ P′ and I ∈ I P, we define

I f : I 7→ ↓f [I] = x′ ∈ P′ | ∃x ∈ I, x′ ≤ f(x),

which is a Scott-continuous function from I P to I P′. It is easy to show that ifL is a ∨-semilattice, then a non-empty lower set I ⊆ L is an ideal iff I is closedunder ∨. Similarly, filters in ∧-semilattices are non-empty upper sets closed under∧.

Basic properties of I and F

The following fact is a consequence of the fact that the definitions of ideals andfilters are order-symmetric. Given a poset P, I is an ideal of Pop iff I is a filter ofP. This gives us an order anti-isomorphism between I(Pop) and F(P).

A.5.2. Fact. I(Pop) ' (F P)op.

We will now list several useful properties of the ideal completion. Naturally,all of these properties dualize to the filter completion.

A.5.3. Fact. Let P be a partial order.

1. I is a functor Pos→ Dcpo and ↓ : Id→ I is a natural transformation.

2. If f : P → D is an order-preserving map to a dcpo D, then there exists aunique Scott-continuous f ′ : I P→ D such that f ′ ↓· = f ; i.e. I P is thefree dcpo over P.

I Pf ′ // D

P

↓P

OO

f

==||||||||

3. If P is a join semilattice then I P is a complete lattice.

4. If P and P′ are join semilattices and if f : P → P′ preserves binary joins,then I f : I P→ I P′ preserves all non-empty joins.

5. If f : P→ Q is an order embedding then so is I f .

A.5. Completions 209

The ideal completion, and dually, the filter completion, preserves finite products;see [1, Prop. 2.2.23].

A.5.4. Fact. Let P,Q be partial orders. There exists an order isomorphismh : I(P×Q)→ I P× I Q such that for all x ∈ P , y ∈ Q,

h ↓P×Q(x, y) = ↓P(x)× ↓Q(y).

(The map h simply sends (I1, I2) to I1×I2.) In other words, the functor I : Pos→Dcpo preserves finite products.

I(P×Q) h // I P× I Q

P×Q

↓P×Q

OO

↓P×↓Q

77ppppppppppp

Moreover, σ↑(I P)× σ↑(I Q) = σ↑(I(P×Q)).

Algebraic dcpo’s

Let D be a dcpo. An element p ∈ D is compact if for all directed S ⊆ D, p ≤∨S

implies that there is some x ∈ S such that p ≤ x. We denote the set of compactelements of D by KD. We call D an algebraic dcpo if for each x ∈ D, the set↓x ∩KD is directed and x =

∨(↓x ∩KD).

A.5.5. Fact. Let D be an algebraic dcpo. Then

1. D ' I(KD);

2. the set ↑ p | p ∈ KD forms a base for the Scott topology on D.

If L is a complete lattice, then KL is closed under finite ∨. Consequently,↓x ∩KL is always directed in a lattice; a complete lattice is therefore algebraiciff for all x ∈ L, x =

∨(↓x ∩KD).

A.5.2 The MacNeille completion of a pre-order

Let P = 〈P,≤〉 be a pre-order and let U ⊆ P . We define

ub(U) := x ∈ P | U ⊆ ↓x (the upper bounds of U),

lb(U) := x ∈ P | U ⊆ ↑x (the lower bounds of U).

A cut of P is a pair of sets (U, V ), where U, V ⊆ P, such that U = lb(V ) andV = ub(U). We can order the set of all cuts of P by setting

(U1, V1) ≤ (U2, V2) :⇔ U1 ⊆ U2 (or equivalently, V1 ⊇ V2).


Under this order, the set of all cuts of P is a complete lattice which we denoteby P. We call P the MacNeille completion of P [70]. There exists a natural mapiP : P→ P defined by

iP : x 7→ (↓x, ↑x).

A.5.6. Remark. Similarly to the canonical extension (Fact 2.1.9), the MacNeillecompletion can be characterized up to isomorphism of completions. Banaschewski& Bruns [9, §4] showed that if e : P→ C is an order embedding of a poset P intoa complete lattice C such that e[P] is both join-dense and meet-dense, then thereexists a unique order-isomorphism h : C→ P such that h e = iP.

A.5.7. Example. Perhaps the best-known example of a MacNeille completionis the embedding of the rational numbers Q into the reals R. This is known asthe completion of the rationals by Dedekind cuts; for this reason the MacNeillecompletion is also known as the Dedekind-MacNeille completion.

A.6 Universal algebra

Our main reference for universal algebra is Burris & Sankappanavar [23].

A.6.1 Ω-algebras

An algebraic signature consists of a set of function symbols Ω and an arity functionar : Ω→ N, assigning a finite arity to each operation symbol. An Ω-algebra is astructure A = 〈A; (ωA)ω∈Ω〉 where for each ω ∈ Ω, ωA is a function ωA : Aar(ω) → A.Given two Ω-algebras A, B, and a function f : A→ B, we say that f is an Ω-algebrahomomorphism if for all ω ∈ Ω, the following diagram commutes:

An

f [n]

ωA // A

f

Bn

ωB // B

where n = ar(ω) and f [n] : An → Bn denotes the function

(x1, . . . , xn) 7→ (f(x1), . . . , f(xn)) .

A.6.2 Homomorphic images, subalgebras and products

Fix an algebraic signature Ω.

A.6. Universal algebra 211

Subalgebras

Let A = 〈A, (ωA)ω∈Ω〉 be an algebra and let B ⊆ A. We say that B is a subuniverseof A if for all ω ∈ Ω, ωA[Bar(ω)] ⊆ B; in other words, if B is closed under alloperations ωA. In that case, we can define operations ωB := (ωA) Bar(ω) for allω ∈ Ω, and we call B = 〈B, (ωB)ω∈Ω〉 a subalgebra of A. If K is a class of algebras,we denote the class of all subalgebras of algebras in K by S(K). If K consists of asingle algebra A, we denote the set of all its subalgebras by S(A).

Homomorphic images, quotients and congruences

Let A and B be algebras and let f : A→ B be a homomorphism; if f is surjectivethen we call B a homomorphic image of A. If K is a class of algebras, we denotethe class of all homomorphic images of algebras in K by H(K). Given an algebra A,each algebra B in the class of all its homomorphic images H(A) can be representedas a quotient, using a congruence on A.

A congruence is an equivalence relation θ ⊆ A×A on the underlying set of Asuch that for all ω ∈ Ω and all a1, . . . , an, b1, . . . , bn ∈ A, where n = ar(ω),

if ai θ bi for all i ≤ n, then also ωA(a1, . . . , an) θ ωA(b1, . . . , bn).

It is a consequence of the definition of congruences that if θ is a congruence ofA = 〈A, (ωA)ω∈Ω〉, then we can define an algebra structure on A/θ, the set of θ-equivalence classes of A. We denote this algebra by A/θ. If we define µθ : A→ A/θas µθ : a 7→ a/θ, then µθ : A → A/θ is a surjective algebra homomorphism. Wedenote the poset of all congruences of A, ordered by subset inclusion, by ConA.In fact, ConA is always a complete lattice, so we may speak of the congruencelattice of A.

A natural example of a congruence is provided by the kernel of a givenhomomorphism f : A→ B:

ker f := (a, b) ∈ A× A | f(a) = f(b).

It is a consequence of the Isomorphism Theorems of universal algebra that iff : A→ B is a surjective homomorphism, then there exists a unique isomorphismh : B → A/ ker f such that h f = µker f . In other words, every homomorphicimage of A is naturally isomorphic to a quotient of A.

Products and subdirect products

If Ai | i ∈ I is a set of algebras, then we can define an algebra structure on theproduct

∏I Ai. If ω is an operation of arity n, then we define ωQ

I Ai on (∏

I Ai)ncoordinate-wise as follows:(

ωQI Ai(a1, . . . , an)

)(i) = (ωAi (a1(i), . . . , an(i))) .


We denote the product of Ai | i ∈ I by∏

I Ai. We denote the correspondingprojection homomorphisms by πj :

∏I Ai → Aj. If K is a class of algebras, we

denote the class of all products of algebras in K by P(K).An algebra A is a subdirect product of a set of algebras Bi | i ∈ I if there

exists a set of surjective algebra homomorphisms pi : A → Bi such that for alla, b ∈ A, if pi(a) = pi(b) for all i ∈ I, then a = b. In this case we call (pi : A→ Bi)Ia subdirect decomposition of A. If K is a class of algebras, we denote the class ofall subdirect products of algebras in K by PS(K).

A.6.3 Varieties

A variety is a class of algebras K which is closed under H, S and P, i.e. such thatH(K) ⊆ K, S(K) ⊆ K and P(K) ⊆ K. It is a fact of universal algebra that if Kis a class of algebras, then HSP(K) is the smallest variety containing K. We callHSP(K) the variety generated by K.

Congruence-distributive varieties

Recall that given an algebra A, we defined ConA to be the lattice of congruencesof A. If ConA is a distributive lattice, we say that A is congruence-distributive.If V is a variety such that every A in V is congruence-distributive, we call V acongruence-distributive variety. The following fact is well-known, cf. [23, Theorem12.3].

A.6.1. Fact. If A has a lattice reduct, then ConA is distributive.

We say that a variety V is finitely generated if there exists a finite algebra Asuch that V = HSP(A). An equivalent condition is to demand that there existsa finite set of finite algebras K such that V = HSP(K). The following fact, forwhich we do not have an explicit reference, is a straightforward consequence ofJonsson’s Lemma [23, Corollary 6.10].

A.6.2. Fact. If V is a congruence-distributive finitely generated variety and ifV ′ is a subvariety of V, then V ′ is also finitely generated.

Proof sketch Suppose that V = HSP(A) for some finite algebra A. Since V iscongruence-distributive, it follows by Jonsson’s Lemma [23, Cor. 6.10] that VSI , thesubdirect irreducibles of V , are contained in HS(A). Consequently, every subdirectirreducible of V is finite. Moreover, since S(A) is finite, HS(A) is essentially finite,meaning that there exists a finite set K ⊆ VSI such that VSI = I(K), whereI(K) is the class of all algebras isomorphic to an algebra in K. Now let V ′ be asubvariety of V; then V ′SI ⊆ VSI , so there must exist a necessarily finite K ′ ⊆ Ksuch that V ′SI = I(K ′). It follows that V ′ = HSP(K ′), so that V ′ is indeed finitelygenerated.

A.6. Universal algebra 213

Boolean products

We recall the definition of Boolean products of algebras from [23, §IV.8]. Let A bean algebra. A Boolean product decomposition of A is a subdirect decomposition ofA, i.e. a collection of surjective maps (px : A→ Bx)x∈X such that for all a, b ∈ A,if px(a) = px(b) for all x ∈ X then a = b, satisfying the additional properties that:

• X is a Boolean space;

• for all a, b ∈ A, the set x ∈ X | px(a) = px(b) is clopen;

• for all a, b ∈ A and all U ⊆ X clopen, there is a (unique) c ∈ A such thatpx(c) = px(a) if x ∈ U and px(c) = px(b) otherwise.

A.6.3. Fact. If A is a finite algebra, then HSP(A) = HSPB(A).

Since this fact seems to be a folklore result, we will sketch the proof below.

Proof sketch First we show that HSP = HSPBPU, where PU stands for takingultraproducts. It is easy to see that HSPBPU ⊆ HSP. For the converse, we willshow that P ⊆ PBPU. Let

∏I Ai be a product of algebras; we will show that∏

I Ai can be seen as the Boolean product of all ultraproducts over Ai | i ∈ I.Define X to be the set of ultrafilters over I; then X is a Boolean space in its Stonetopology. For each U ∈ X, let pU :

∏I Ai →

∏I Ai/U be the projection onto the

ultraproduct∏

I Ai/U . We claim that(pU :

∏IAi →

∏IAi/U

)U∈X

is a Boolean product decomposition. It is not hard to show the following facts:

1. (pU :∏

I Ai →∏

I Ai/U)X is a subdirect product decomposition: if a, b ∈∏I Ai and a 6= b, then there exists i ∈ I such that a(i) 6= b(i). If we take the

principal ultrafilter U := J ⊆ I | i ∈ J then we see that pU(a) 6= pU(b).

2. For all a, b ∈∏

I Ai, U ∈ X | pU(a) = pU(b) is clopen: it follows fromthe definition of pU that pU(a) = pU(b) iff i ∈ I | a(i) = b(i) ∈ U , andU ∈ X | i ∈ I | a(i) = b(i) ∈ U

is clopen by Stone duality.

3. For all a, b ∈∏

I Ai and all clopen O ⊆ X, there exists a c ∈∏

I Ai suchthat pU(c) = pU(a) if U ∈ O and pU(c) = pU(b) if U ∈ X \O: since O ⊆ Xis clopen, by Stone duality there exists an J ⊆ I such that U ∈ O iff J ∈ U .Now define c(i) := a(i) if i ∈ J and c(i) := b(i) if i ∈ I \ J .

Since Ai | i ∈ I was arbitrary, it follows that P ⊆ PBPU, so that indeedHSP = HSPBPU. Returning to the statement of the Fact, if A is finite, thenPU(A) = I(A); it follows that HSP(A) = HSPB(A).


A.6.4 Terms and equations

Let X be a set of variables. The set TerΩ(X) of Ω-terms over X is defined as thesmallest set such that

• X ⊆ TerΩ(X);

• for all ω ∈ Ω, for all t1, . . . , tar(ω) ∈ TerΩ(X), we have ω(t1, . . . , tar(ω)) ∈TerΩ(X).

If t is a term, we write t(x1, . . . , xn) to indicate that the variables occuring in tare among x1, . . . , xn. If t(x1, . . . , xn) is an Ω-term and A is an Ω-algebra, thenwe define the term function tA : An → A inductively as

(xi)A := πi : An → A,(ω(t1(x), . . . , tk(x))A := ωA ((t1)A × · · · × (tk)A) ,

where x denotes the tuple x1, . . . , xn and k = ar(ω). If s(x1, . . . , xn) andt(x1, . . . , xn) are terms we say that the equation s ≈ t is valid on an algebraA (alternatively, A satisfies s ≈ t or A |= s ≈ t) if sA = tA, i.e. if the correspondingterm functions coincide. If A is an ordered algebra, then A satisfies the inequations 4 t if sA ≤ tA.

A.7 General topology

Our main reference for general topology is Engelking [33]. Another standard textis Bourbaki [22].

A.7.1 Topological spaces

Let X be a set. A topology on X is a collection of subsets τ ⊆ P(X), which wecall τ -open sets, such that

• ∅, X ⊆ τ ;

• for all S ⊆ τ ,⋃S ∈ τ ;

• for all U, V ∈ τ , U ∩ V ∈ τ .

We call 〈X, τ〉 a topological space. If it is clear what the topology on X is wewill not mention τ explicitly. We define U ⊆ X to be closed if X \ U is open,i.e. if X \ U ∈ τ . If both U and X \ U are open, then we say U is clopen(closed-and-open).

A.7. General topology 215

A.7.1. Example. Given any set X, we can always define two somewhat trivialtopologies on it. The space 〈X, ∅, X〉 is literally called the trivial topology; thistopology has the smallest collection of open sets possible. On the other extremewe have the discrete topology 〈X,P(X)〉, which is the topology on X with thelargest possible collection of open sets.

Subspaces

If 〈X, τ〉 is a topological space and Y ⊆ X is a set, we can define a topology onY , called the subspace topology , by defining

τY := U ∩ Y | U ∈ τX.

We then call 〈Y, τY 〉 a subspace of 〈X, τX〉.

Bases and subbases

Let X be a set. A subbase for a topology on X is a collection B ⊆ P(X) such that

•⋃B = X.

We call B a base if addionally,

• ∀U, V ∈ B, U ∩ V =⋃W ∈ B | W ⊆ U, V .

If B is a subbase on X, then ⋂S | S ⊆ B finite is a base on X.

A.7.2. Example. As a trivial example, observe that if 〈X, τ〉 is a topologicalspace, then τ is a base on X.

Let B be a base on X and let B′ be a subbase on X. We define

〈B〉 = ⋃B0 | B0 ⊆ B, and

〈B′〉 =⋃B0 | B0 ⊆

⋂S | S ⊆ B finite

.

It is a fact of general topology that 〈B〉 and 〈B′〉 as defined above are topologieson X. Given a topological space 〈X, τ〉, we say B is a (sub-) base for 〈X, τ〉 ifτ = 〈B〉. If B is a base for 〈X, τ〉, then it follows from the definition of 〈B〉 that forall open sets U ∈ τ and for all x ∈ U , there exists a V ∈ B such that x ∈ V ⊆ U .

Let X be a set. One can show that the partial order⟨τ ⊆ P(X) | 〈X, τ〉 is a topological space,⊆

⟩is a complete lattice. If τ0 and τ1 are topologies on X with bases B0 and B1,respectively, then

U ∩ V | U ∈ B0, V ∈ B1is a base for τ0 ∨ τ1.


A.7.2 Continuity

Let 〈X, τX〉 and 〈Y, τY 〉 be topological spaces. A function f : X → Y is called(τX , τY )-continuous if for all U ∈ τY , f−1(U) ∈ τX . If it is clear from the contextwhat the topologies are, we simply say that f is continuous. We say that f : X → Yis (τX , τY )-open if for all U ∈ τX , f [U ] ∈ τY . We say that 〈X, τX〉 and 〈Y, τY 〉 arehomeomorphic if there exists an open and continuous bijective map f : X → Y ;this is the natural notion of isomorphism for topological spaces.

Let f : Y → X be a continuous function between spaces 〈X, τX〉 and 〈Y, τY 〉.If f is injective and if for all U ∈ τY , there exists a U ′ ∈ τX such that U = f−1(U ′),then we call f : X → Y a homeomorphic embedding. Under these circumstances,〈Y, τY 〉 is homeomorphic to f [Y ], where the latter set is equipped with the subspacetopology inherited from 〈X, τX〉.

One of the reasons it is practical to work with bases and subbases for topologiesis that they make it easier to check whether a function f : X → Y betweentopological spaces 〈X, τX〉 and 〈Y, τY 〉 is continuous: If B is a subbase for τY , thenone can show that f : X → Y is continuous iff for all U ∈ B, f−1(U) ∈ τX .

A.7.3. Lemma. Let X, Y be sets, let σ0, σ1 be topologies on X, let τ0, τ1 betopologies on Y and let f : X → Y be a function.

1. If f : X → Y is (σ0, τ0)-continuous and if σ0 ⊆ σ1 and τ0 ⊇ τ1, then f isalso (σ1, τ1)-continuous.

2. If f : X → Y is both (σ0, τ0)-continuous and (σ1, τ1)-continuous, then f isalso (σ0 ∨ σ1, τ0 ∨ τ1)-continuous.

Proof This is an easy exercise.

A.7.3 Separation and compactness

In this subsection we list several properties of topological spaces. A topologicalspace 〈X, τ〉 satisfies the T0 separation axiom if for all x, y ∈ X such that x 6= y,there exists an open set U ⊆ X such that either x ∈ U 63 y or x /∈ U 3 y. Atopological space satisfies the T1 separation axiom if for all x, y ∈ X such thatx 6= y, there exist open sets U, V ⊆ X such that x ∈ U 63 y and x /∈ V 3 y.Finally, a topological space satisfies the T2 separation axiom, also known as theHausdorff separation axiom, if for all x, y ∈ X such that x 6= y, there exist disjointopen sets U, V ⊆ X such that x ∈ U and y ∈ V . We will usually simply say thata space is T0, Hausdorff, etc.

Let 〈X, τ〉 be a topological space and let Y ⊆ X. We say Y is compact if forall S ⊆ τ such that Y ⊆

⋃S, there exists a finite S0 ⊆ S such that Y ⊆

⋃S0. If

X itself is a compact set we call 〈X, τ〉 a compact space.

A.8. Duality for ordered Kripke frames 217

Boolean spaces

A topological space 〈X, τ〉 is zero-dimensional if 〈X, τ〉 has a base of clopen sets.Equivalently, 〈X, τ〉 is zero-dimensional if for all open sets U ⊆ X and all x ∈ U ,there exists a clopen V ⊆ X such that x ∈ V ⊆ U . We call spaces that are bothcompact, Hausdorff and zero-dimensional Boolean spaces; these spaces are alsocalled Stone spaces [54].

A.8 Duality for ordered Kripke frames

In §4.1.4, we discuss the discrete duality between DLO+, the category of semi-topological distributive lattices with operators and complete homomorphisms, andOKFr, the category of ordered Kripke frames and bounded morphisms. However,in §4.1.4 we only describe the functor (·)+ : OKFrop → DLO+. Below we willdescribe the functor (·)+ : DLO+ → OKFrop. As a general reference about DLO’swe mention Goldblatt [47]; the details of the discrete duality we use are discussedby Gehrke et al. in [40, §2.3].

Before we can describe the functor (·)+ : DLO+ → OKFrop, we need a fewdefinitions. Let A be a semi-topological DLO. By J∞(A) we denote the completelyjoin-irreducible elements of A, i.e. those p ∈ A such that for all S ⊆ A, if p =

∨S

then there exists an x ∈ S such that p = x. Using order duality we can definethe set of completely meet irreducible elements of A, denoted M∞(A). In a semi-topological DLO, we can define an order-isomorphism κ : J∞(A)→ M∞(A), bysending p 7→

∨(A \ ↑ p).

Recall from §4.1.1 that a semi-topological DLO is a complete bi-algebraicdistributive lattice-based algebra A = 〈A,∧A,∨A, 0A, 1A,♦A,A〉, where ♦A : An →A is a complete normal n-ary operator and A : Am → A is a complete normalm-ary dual operator. We now define

A+ := 〈J∞(A),≤, RA+ , QA+〉,

where

• 〈J∞(A),≤〉 is the set of completely join-irreducibles of A with the orderinherited from A;

• RA+ is an n+ 1-ary relation on J∞(A) such that

p RA+ (q1, . . . , qn) iff p ≤ ♦A(q1, . . . , qn);

• QA+ is an m+ 1-ary relation on J∞(A) such that

p QA+ (q1, . . . , qm) iff A(κ(q1), . . . , κ(qm)) ≤ κ(p).


If f : A→ B is a morphism of DLO+, i.e. a complete DLO homomorphism, thenwe define f+ : B+ → A+ as

f+ : p 7→∧a ∈ A | p ≤ f(a),

which is indeed a bounded morphism. One can show that f+ is in fact the leftadjoint of f : A→ B, restricted to J∞(B).

A.8.1. Fact. The functors (·)+ : DLO+ → OKFrop and (·)+ : OKFrop → DLO+

form a dual equivalence of categories. Each of the functors (·)+ and (·)+ mapfinite objects to finite objects. Moreover given a complete DLO homomorphismf : A→ B between semi-topological DLO’s,

1. f : A→ B is surjective iff f+ : B+ → A+ is an embedding of ordered Kripkeframes;

2. f : A→ B is injective iff f+ : B+ → A+ is surjective.

The isomorphisms witnessing the above Fact are obtained as follows: for asemi-topological DLO A, we map A to (A+)+ by sending

a 7→ p ∈ J∞(A) | p ≤ a,

which is a lower set of A+. Conversely, given an ordered Kripke frame F, we mapF to (F+)+ by sending

x 7→ ↑x,

which is a completely join-irreducible element of F+.

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Index

adjunctionof categories, 202of posets, 204

algebra, 210Boolean product, 213congruence, 211homomorphic image, 211product, 211quotient, 211subalgebra, 211subdirect product, 211, 212

altn, 136

base for a topology, 215Base of a functor, 147, 150binary tree functor, 147Boolean algebra, 129, 207Boolean algebra with operators (BAO),

133compact Hausdorff, 135profinite, 136semi-topological, 133

Boolean topologicalalgebra, 60BAO, 135Boolean algebra, 129DLO, 115lattice, 65lattice-based algebra, 105space, see topology, Boolean

bounded morphism, 116

canonical extension type, 86canonical extension of a lattice

definition, 13fixed points, 14functor, 43non-continuous, 97products, 24topological characterization, 21via dcpo presentations, 48when profinite, 77

canonical extension of a lattice-basedalgebra

definition, 86finite, 87functor, 87universal property, 94, 105when profinite, 99

canonical extension of a mapalgebra homomorphism, 87composition, 42, 78continuity, 31, 40, 74definition

arbitrary map, 73order-preserving map, 29

maximality, 31, 77order embedding, 32via dcpo presentations, 50via lim inf, 69

227

228 Index

canonicity, 89category, 200

dual, 201cocone, 117

map of cocones, 117colimit, 117compact Hausdorff

algebra, 55BAO, 135Boolean algebra, 129lattice, 63

compactification, 55, 136completely join-irreducible, 26, 217completion, 13, 207complex algebra, 123composition of relations vs functions,

144cone, 57

map of cones, 57congruence, 211congruence lattice, 211

dcpo, 204algebra, 90algebraic, 209presentation, 47

diagram, 57associated, 120directed, 117

directed set, 204distributive lattice, 41, 128, 206distributive lattice with operators (DLO),

113Boolean topological, 115profinite, 115, 121semi-topological, 114

dualityBAO’s, 133, 135Boolean algebras, 129distributive lattices, 128DLO’s, 124, 217Heyting algebras, 130profinite DLO’s, 120

topological BAO’s, 132

equation, 214equivalence of categories, 202

filter, 208completion, 208

filter elements, 14finitely generated variety, 77, 87, 89,

99, 212flat site, 153frame (lattice), 152

compact, 155presentation, 152

functor, 201finitary, 146standard, 146weak pullback preserving, 145

examples, 146

generated subframe, see Kripke frame

Heyting algebra, 130, 207homeomorphic embedding, 216homomorphism, 210

ideal, 208completion, 208

ideal elements, 14insertion of generators, 153

Kripke framedefinition, 116, 133duality, 217generated subframe, 119hereditarily finite, 121image-finite, 133intuitionistic, 130

lattice, 206lattice reduct, 87lattice-based algebra, 84lifted members of a set, 151limit, 58limiting cone, 58

Index 229

lower set, 203

MacNeille completion, 128, 209

natural transformation, 201normal operator, 37

operator, 37order signature, 85order type, 85order-preserving map, 203ordered Kripke frame, see Kripke frame

partially ordered set, 203poset, see partially ordered setpower set monad, 145power set functor, 199pre-order, 202prime filter frame, 123profinite

BAO, 136DLO, 115, 121

profinite algebra, 58profinite completion, 62, 94, 126

when equal to canonical extension,99

pullback, 145

relation lifting, 147examples, 148

Scott-continuity, 204semilattice, 205slim redistribution, 158smooth

lattice-based algebra, 86map, 39, 78

span, 148Stone space, see topology, Booleansubbase, 215subcategory, 200

co-reflective, 125, 202reflective, 202

substructure, 119suplattice, 153, 206

presentation, 153

T -powerlocalecompactness, 191definition, 159functor, 171natural transformations, 173regularity, 185via flat sites, 179zero-dimensionality, 185

topological algebra, 54topology, 214

and order, 205Boolean, 217compactness, 216continuous function, 216δ-, 16, 21interval, 15Lawson, 65, 67Scott, 15, 205separation axioms, 216subspace, 215zero-dimensional, 217

ultrafilter extensionembedding into, 138

ultrafilter frame, see prime filter frame,137

universal algebra, 210universal property, 94, 105upper set, 203

variety, 212congruence-distributive, 212finitely generated, see finitely gen-

erated varietyVietoris

hyperspace, 141powerlocale, 154, 165

weak pullback, 145

Samenvatting

In dit proefschrift bespreken we drie onderwerpen: canonieke extensies van tralie-algebra’s, Stone-dualiteit voor distributive tralies met operatoren, en een generali-satie van de puntvrije Vietoris powerlocale-constructie.

In Hoofdstukken 2 en 3 onderzoeken verbanden tussen canonieke extensiesvan tralie-algebra’s en topologische algebra, pro-eindige completeringen en gericht-volledige partiele ordeningen (dcpo’s). We geven een topologische karakteriseringvan de canonieke extensie van een tralie in §2.1.3, en een verbeterde karakteriseringvan canonieke extensies van orde-bewarende afbeeldingen als maximale continueextensies, alsmede andere continuıteitsresultaten, in §2.2. De verbetering in deresultaten in §2.2 schuilt in het feit dat we niet uitgaan van distributieve tralies,maar van willekeurige tralies. In §2.3 ronden we Hoofdstuk 2 af met een andere ka-rakterisering van canonieke extensies, namelijk door middel van dcpo-presentaties.In Hoofdstuk 3 bespreken we canonieke extensies van willekeurige afbeeldingen encanonieke extensies van algebras, beide in relatie tot topologische algebra. In §3.3.2laten we zien dat canonieke extensies van surjectieve algebrahomomorfismes tussentralie-algebra’s wederom homomorfismes zijn. We gebruiken dit gegeven in §3.4.1om te laten zien dat de pro-eindige completering van een willekeurige tralie-algebraA gekarakteriseerd kan worden als een volledig quotient van de canonieke extensievan A. Vervolgens onderzoeken we in §3.4.2 wanneer de pro-eindige completeringvan een tralie-algebra A samenvalt met de canonieke extensie van A. We sluitenHoofdstuk 3 af met enige resultaten betreffende een universele eigenschap vancanonieke extensies met betrekking tot Boolese topologische algebra’s in §3.4.3.

In Hoofdstuk 4 verdiepen wij ons in de discrete Stone-dualiteit voor semi-topologische distributieve tralies met operatoren (DLO’s) en geordende Kripke-frames. In §4.1 behandelen we de dualiteit tussen pro-eindige DLO’s en dedaarmee corresponderende erfelijk eindige geordende Kripke-frames. We besprekenenkele speciale gevallen van deze dualiteit in §4.2, te weten discrete dualiteit voordistributieve tralies, Boolese algebra’s, Heytingalgebra’s en modale algebra’s.Wij sluiten dit hoofdstuk af met een nadere blik op de voornoemde dualiteit in het

231

232 Samenvatting

geval van Boolese algebra’s met operatoren (BAO’s) in §4.3. In dit geval kunnenwe niet alleen de pro-eindige BAO’s karakteriseren middels Stone-dualiteit, maarook de compacte Hausdorff- en Boolese topologische BAO’s. We laten zien datcompacte Hausdorff-BAO’s (en Boolese topologische BAO’s) de dualen zijn vaneindig vertakkende Kripke-frames. We passen deze kennis toe in een beschouwingvan de inbedding van Kripke-frames in hun ultrafilterextensies in §4.3.2.

In Hoofdstuk 5 gebruiken we een geometrische variant van de Carioca-axioma’svoor coalgebraısche modale logica om een nieuwe beschrijving te geven vande puntvrije Vietorisconstructie. In §5.3.1 introduceren we de T -powerlocale-constructie, gegeven een endofunctor T : Set → Set op de categorie van verza-melingen. We laten vervolgens in §5.3.3 zien dat het P -powerlocale, waar P decovariante machtsverzamelingfunctor is, de oorspronkelijke Vietoris powerlocale-constructie geeft. In §5.3.4 laten we zien dat de T -powerlocale-constructie onder-deel is van een functor VT op de categorie van frames. Daarnaast laten we zienhoe een natuurlijke transformatie van een functor T ′ naar T een natuurlijke trans-formatie van VT naar VT ′ oplevert. In §5.3.5 laten we zien dat het T -powerlocalegepresenteerd kan worden door middel van flat sites; dit leidt tot een algebraıschbewijs van het feit dat formules in onze geometrische coalgebraısche modale logicadisjunctieve normaalvormen hebben. Ten slotte laten we in §5.4 zien dat de T -powerlocale-constructie frame-eigenschappen als regulariteit, nul-dimensionaliteiten de combinatie van nul-dimensionaliteit en compactheid bewaart.

Abstract

In this dissertation we discuss three subjects: canonical extensions of lattice-basedalgebras, Stone duality for distrbutive lattices with operators, and a generalizationof the point-free Vietoris powerlocale construction.

In Chapters 2 and 3, we study canonical extensions of lattice-based algebrasin relation to topological algebra, profinite completions and directed completepartial orders (dcpo’s). We provide a topological characterization theorem for thecanonical extension of a lattice in §2.1.3, and we give an improved characterizationof canonical extensions of order-preserving maps as maximal continuous extensions,along with further continuitresults, in §2.2. The improvement in the results of§2.2 lies in the fact that they hold for arbitrary rather than distributive lattices.In §2.3, we show how canonical extensions of lattices can be characterized usingdcpo presentations, concluding Chapter 2. In Chapter 3 we discuss canonicalextensions of arbitrary maps and canonical extensions of lattice-based algebras,both in relation to topological algebra. In §3.3.2, we show that the canonicalextenion of a surjective lattice-based algebra homomorphism is again an algebrahomomorphism. We use this fact to show in §3.4.1 that the profinite completionof any lattice-based algebra A can be characterized as a complete quotient ofthe canonical extension of A. Subsequently, in §3.4.2, we investigate necessaryand sufficient circumstances for the profinite completion of A to be equal to thecanonical extension of A. We conclude Chapter 3 with a discussion of a universalproperty of canonical extensions with respect to Boolean topological algebras in§3.4.3.

In Chapter 4, we study discrete Stone duality for semi-topological distributivelattices with operators (DLO’s) and ordered Kripke frames. In §4.1, we study theduality between profinite DLO’s and the corresponding hereditarily finite orderedKripke frames. We consider special cases of this duality in §4.2, namely distributivelattices, Boolean algebras, Heyting algebras and modal algebras. Finally, in §4.3,we show that if we restrict our attention to Boolean algebras with operators(BAO’s) rather than DLO’s, we can characterize not only profinite BAO’s via

233

234 Abstract

Stone duality, but also compact Hausdorff and Boolean topological BAO’s. Weshow that compact Hausdorff BAO’s (and Boolean topological BAO’s) are dualto image-finite Kripke frames. We use this knowledge to study the embedding ofKripke frames into their ultrafilter extensions in §4.3.2.

In Chapter 5, we use a geometric version of the Carioca axioms for coalgebraicmodal logic with the cover modality to give a new description of the point-freeVietoris construction. In §5.3.1 we introduce the T -powerlocale construction,where T : Set→ Set is a weak-pullback preserving, standard, finitary endofunctoron the category of sets. We then go on to show that the P -powerlocale, whereP is the covariant powerset functor, is the usual Vietoris powerlocale in §5.3.3.In §5.3.4 we show that the T -powerlocale construction yields a functor VT on thecategory of frames, and we show how to lift natural transformations between setfunctors T ′ and T to natural transformations between T -powerlocale functors VTand VT ′ . In §5.3.5 we show that the T -powerlocale can be presented using a flatsite presentation rather than an frame presentation; this gives us an algebraicproof for the fact that formulas in our geometric coalgebraic modal logic havea disjunctive normal form. Finally in §5.4, we show that the T -powerlocaleconstruction preserves regularity, zero-dimensionality and the combination ofzero-dimensionality and compactness.

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ILLC DS-2006-04: Robert SpalekQuantum Algorithms, Lower Bounds, and Time-Space Tradeoffs

ILLC DS-2006-05: Aline HoninghThe Origin and Well-Formedness of Tonal Pitch Structures

ILLC DS-2006-06: Merlijn SevensterBranches of imperfect information: logic, games, and computation

ILLC DS-2006-07: Marie NilsenovaRises and Falls. Studies in the Semantics and Pragmatics of Intonation

ILLC DS-2006-08: Darko SarenacProducts of Topological Modal Logics

ILLC DS-2007-01: Rudi CilibrasiStatistical Inference Through Data Compression

ILLC DS-2007-02: Neta SpiroWhat contributes to the perception of musical phrases in western classicalmusic?

ILLC DS-2007-03: Darrin HindsillIt’s a Process and an Event: Perspectives in Event Semantics

ILLC DS-2007-04: Katrin SchulzMinimal Models in Semantics and Pragmatics: Free Choice, Exhaustivity, andConditionals

ILLC DS-2007-05: Yoav SeginerLearning Syntactic Structure

ILLC DS-2008-01: Stephanie WehnerCryptography in a Quantum World

ILLC DS-2008-02: Fenrong LiuChanging for the Better: Preference Dynamics and Agent Diversity

ILLC DS-2008-03: Olivier RoyThinking before Acting: Intentions, Logic, Rational Choice

ILLC DS-2008-04: Patrick GirardModal Logic for Belief and Preference Change

ILLC DS-2008-05: Erik RietveldUnreflective Action: A Philosophical Contribution to Integrative Neuroscience

ILLC DS-2008-06: Falk UngerNoise in Quantum and Classical Computation and Non-locality

ILLC DS-2008-07: Steven de RooijMinimum Description Length Model Selection: Problems and Extensions

ILLC DS-2008-08: Fabrice NauzeModality in Typological Perspective

ILLC DS-2008-09: Floris RoelofsenAnaphora Resolved

ILLC DS-2008-10: Marian CounihanLooking for logic in all the wrong places: an investigation of language, literacyand logic in reasoning

ILLC DS-2009-01: Jakub SzymanikQuantifiers in TIME and SPACE. Computational Complexity of GeneralizedQuantifiers in Natural Language

ILLC DS-2009-02: Hartmut FitzNeural Syntax

ILLC DS-2009-03: Brian Thomas SemmesA Game for the Borel Functions

ILLC DS-2009-04: Sara L. UckelmanModalities in Medieval Logic

ILLC DS-2009-05: Andreas WitzelKnowledge and Games: Theory and Implementation

ILLC DS-2009-06: Chantal BaxSubjectivity after Wittgenstein. Wittgenstein’s embodied and embedded subjectand the debate about the death of man.

ILLC DS-2009-07: Kata BaloghTheme with Variations. A Context-based Analysis of Focus

ILLC DS-2009-08: Tomohiro HoshiEpistemic Dynamics and Protocol Information

ILLC DS-2009-09: Olivia LadinigTemporal expectations and their violations

ILLC DS-2009-10: Tikitu de Jager“Now that you mention it, I wonder. . . ”: Awareness, Attention, Assumption

ILLC DS-2009-11: Michael FrankeSignal to Act: Game Theory in Pragmatics

ILLC DS-2009-12: Joel UckelmanMore Than the Sum of Its Parts: Compact Preference Representation OverCombinatorial Domains

ILLC DS-2009-13: Stefan BoldCardinals as Ultrapowers. A Canonical Measure Analysis under the Axiom ofDeterminacy.

ILLC DS-2010-01: Reut TsarfatyRelational-Realizational Parsing

ILLC DS-2010-02: Jonathan ZvesperPlaying with Information

ILLC DS-2010-03: Cedric DegremontThe Temporal Mind. Observations on the logic of belief change in interactivesystems

ILLC DS-2010-04: Daisuke IkegamiGames in Set Theory and Logic

ILLC DS-2010-05: Jarmo KontinenCoherence and Complexity in Fragments of Dependence Logic

ILLC DS-2010-06: Yanjing WangEpistemic Modelling and Protocol Dynamics

ILLC DS-2010-07: Marc StaudacherUse theories of meaning between conventions and social norms

ILLC DS-2010-08: Amelie GheerbrantFixed-Point Logics on Trees

ILLC DS-2010-09: Gaelle FontaineModal Fixpoint Logic: Some Model Theoretic Questions

ILLC DS-2010-10: Jacob VosmaerLogic, Algebra and Topology. Investigations into canonical extensions, dualitytheory and point-free topology.

logic, algebra and topology · 2010-11-15 · logic, algebra and topology investigations into...

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